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\vspace*{12pt}

{\large {\bf Bertrand Competition Under Uncertainty} \newline
}

\begin{center}
\bigskip June 5, 2001 \\[0pt]
\bigskip

Maarten Janssen and Eric Rasmusen \\[0pt]
\bigskip

{\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 activity level
can be endogenized in several ways-- whether to incur a fixed cost of
activity, for example, or what level of output to choose. Our 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 decline more sharply than in the Cournot model, and
the pattern is similar to that found empirically by Bresnahan and Reiss
(1991).

{\small \noindent \hspace*{20pt} Janssen: Department of Economics, H7-22,
Erasmus University, 3000 DR Rotterdam, The Netherlands. Fax: 31-10-4081949. 
Janssen@few.eur.nl. }

{\small \hspace*{20pt} Rasmusen: Visiting Professor, CIRJE, Dept. of
Economics, University of Tokyo (2001), Professor of Business Economics and
Public Policy and Sanjay Subhedar Faculty Fellow, Indiana University, Kelley
School of Business, BU 456, 1309 East 10th Street,  Bloomington, Indiana,
47405-1701. (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu. Http:
//Php.indiana.edu/$\sim$erasmuse. Copies of this paper can be found at%
\newline
Php.indiana.edu/$\sim$erasmuse/papers/bertrand.pdf. }

{\small We thank David Schmidt, two anonymous referees, and participants in
seminars at the Indiana University Dept. of Economics, Erasmus University,
and CIRANO in Montreal for their comments. Rasmusen thanks Harvard Law
School's Olin Center and the University of Tokyo's Center for International
Research on the Japanese Economy for their hospitality. }

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

\newpage

\noindent {\it 1. Introduction}

Consider a carpenter who is asked by a homeowner to submit a tender for
renovating a house. He considers it very likely that if the homeowner has
asked for tenders from other carpenters then the lowest price will win the
job. He also knows there is a chance that the homeowner has not found any
other carpenter free to do the work this month and will give the job to him
even if his tender is rather high. What price will the carpenter offer the
homeowner?

The price will certainly be above marginal cost. The carpenter knows that
with some probability he is a monopolist who can charge the monopoly price,
even though with some probability he does face competition. We will model
the situation and show that there exists an equilibrium in mixed strategies
and that expected industry profits are positive for any number of firms.
Moreover, not only do expected profits rise with seller concentration, but
the model does reasonably well in explaining the empirical results of
Bresnahan and Reiss (1991) on {\it how} industry profits increase.

The model allows for a number of interpretations. First, uncertainty about
the existence of competitors may arise from uncertainty on the demand side,
with respect to consumer information. It may be unclear whether consumers
regard rival commodities as perfect substitutes, consumer search costs may
be uncertain from the firm's perspective, or consumers may vary in their
sophistication. Examples of this range from grocery shopping to buying
clothing from mail order companies depending on what catalogs have been
received to buying beers that to some consumers all taste alike but to other
consumers do not. Any of these things might result in a given consumer not
knowing the prices every firm is charging.

Second, uncertainty about the vigor of competition may arise from
uncertainty about the supply side. It may be unclear whether rivals have hit
their capacity constraints (in which case they cannot compete for additional
consumers), whether rivals have entered yet, whether rivals have grossly
overpriced by mistake or ignorance, or whether rivals have temporarily high
costs. It may be unclear whether other competitors have also discovered a
new market, or in black markets it may be difficult to know the number of
firms operating in that market (cf. Janssen and Van Reeven, 1998). Examples
of these range from wholesale distribution of candy bars (where in periods
of peak demand first one and then another manufacturer may hit capacity) to
airline ticket pricing to sales of unusual but not rare antiques or used
books. Any of these situations can be modelled as uncertainty over the
number of active rivals.

The demand and supply interpretations of the previous two paragraphs allow
us to treat the probability a firm is actively competing for a given
customer as an exogenous probability, independent of the number of
potentially active firms. This probability is beyond a firm's control and
can be regarded in Bayesian fashion as a decision by Nature. We also will
show how for a given number of firms the probability of being active can
arise from a firm's decision taken in response to uncertainty about previous
decisions by other firms.  The two-stage models in Section 3 will endogenize
the probability that a given number of firms are active in competing for a
given consumer. We will show how in two different settings entry is a random
decision in equilibrium, in an auction-like setting, as in the carpenter
example, and when firms set output or capacity An important feature of both
settings will be that a firm's decision whether to compete for a given
consumer is not observed before other firms decide their prices.  A
disadvantage of endogenizing the probability of inactivity, however, is that
the probability will change with the number of potentially active firms and
the size of market demand. Thus, when we come to compare our results with
the empirical findings of Bresnahan and Reiss, it is the exogenous
interpretation of firm activity that we will apply.

The paper is related to several different literatures. A variety of models,
of which Salop and Stiglitz (1977) and Varian (1980) are early examples,
have shown that competitive markets can have price dispersion even in
equilibrium. Different firms charge different prices for an identical good
because of heterogeneous consumer search, some consumers observing more
prices than others. The closest of these to the present model is Burdett and
Judd (1983), in which some consumers might observe one price, some two
prices, some three, and so forth. The number of searches is endogenous, and
in equilibrium a given consumer observes only one or two prices.  Our model
differs in a number of respects. First, while one way to look at our model
is that consumers differ with respect to the number of prices they observe,
our model allows for the other interpretations mentioned earlier,
interpretations inappropriate for search models. Indeed, since since the
number of prices observed by a consumer is exogenous in our model,
endogenous search is not a good interpretation. Second, the firms in our
model are strategic, not competitive. This allows us to  study the impact on
pricing behavior of the number of firms, a variable not relevant in Burdett
and Judd (1983). Finally and most important, we treat uncertainty
differently. In our model, a firm believes there is a fixed probability that
any of its competitors is active, whereas in Burdett and Judd it is the
probability that a consumer observes a certain number of prices that is
exogenous. The difference lies in what happens as the number of sellers
increases. In our model, a seller knows that the probability that at least
one other firm is actively competing with it has become closer to one. This
drives prices closer to marginal cost, and in the limit we obtain the
standard Bertrand outcome. Burdett and Judd still have price dispersion and
positive industry profits as the number of firms becomes infinite, because
each firm may still be visited by a non-negligible number of consumers who
do not search for other prices. In their model, the reason a consumer pays a
high price is not that low prices are not available, but that he does not
know where to find them.

Also related is Elberfeld and Wolfstetter (1999). They consider a two-stage
model in which firms first decide whether to enter and then compete in
prices. The outcome of the first stage is known before the firms set their
prices in the second stage. Thus, the outcome in the second stage is
standard: a firm charges the monopoly price if it is the only firm in the
market, otherwise prices are equal to marginal cost. Their main result is
that the probability that no firm enters the market increases with the
number of potential competitors. Their analysis is closely related to our
two-stage game, the important difference being that in our model the entry
decision itself is not oberved before firms compete in prices.

Spulber (1995) analyzes a model of Bertrand competition when firms' cost
functions are private information. He shows that the model has a unique pure
strategy equilibrium in which firms set prices above marginal cost and have
positive expected profits. In contrast, the firms in our model do not know
how many competitors they have, but assume that any competitor that does
exist has the same cost structure. Even though the type of uncertainty
varies between Spulber's model and ours, the properties of the market
equilibrium are similar: firms set prices above marginal cost and receive
positive expected profits.

Another approach to positive profits under Bertrand competition can be found
in the epsilon-equilibrium Bertrand model of Baye and Morgan (1999). They
show that if firms only choose prices to reach within epsilon of their
maximal profit, then a mixed-strategy equilibrium exists, in which profits
are positive and large compared to the value of epsilon. Thus, if, due to
satisficing or managerial slack, firms do not maximize profits completely,
the Bertrand model generates more realistic outcomes. The model in our paper
also introduces noise which generates a mixed strategy equilibrium, but our
noise is the possibility that a customer does not have alternative sellers
from whom to buy.

Finally, our model is also of interest for students of auctions. The
similarities between Bertrand price competition and first-price sealed-bid
auctions is well-known, as, e.g., Baye and Morgan (1997a, b) explain. Our
paper can be regarded as answering the question what is the optimal bid if
the number of participantsin a sealed-bid auction is unknown, as is often
the case in procurement bids, adding to the literature of which McAfee and
Macmillan (1987) is an example.

Section 2 of the paper lays out the basic model and solves for the mixed
strategy equilibrium. Section 3 shows how the entry decision can be
endogenized in three different types of two-stage games. These three models
also show some of the alternative ways our basic model may be interpretated.
Section 4 compares the outcome in the model with that of a Cournot model,
and compares the expected industry profits in our model for different
numbers of potentially active firms with the empirical findings of Bresnahan
and Reiss (1991). Section 5 concludes.

\bigskip

\noindent {\it 2. The Model}

Let there be $N$ firms that might produce a homogeneous good. Before
deciding price, a firm does not know how many other firms are active in
market. The probability a given firm is active is $\alpha$, where $0 \leq
\alpha \leq 1$. If $\alpha=1$, the market is described by the Bertrand model
of price competition, and the equilibrium price equals marginal cost. If $%
\alpha=0$ so our one firm is assured of being a monopolist, it will charge
the monopoly price. For simplicity, we will assume that there is one
consumer, who buys at most one unit, and his maximum willingness to pay for
the good is $v$. In case of tied prices, the consumer picks a firm randomly.
Marginal cost is normalized to 0.

First, let us establish that there is no symmetric Nash equilibrium with any
firm putting positive probability on choosing any particular price on the
continuum. Suppose Firm 1 (without loss of generality) charges price $%
p^{\prime}$ with positive probability $\theta$, rather than mixing over a
continuous range of prices and putting infinitesimal probability on each.
Putting positive probability on $p^{\prime}=0$ is not profit maximizing,
because if the firm charged the monopoly price of $v$ instead on those
occasions it would have an expected payoff of $(1-\alpha)^{N-1} v$, so let
us focus on $p^{\prime}>0$.

If $p^{\prime}>0$, and two firms are putting positive probability $\theta$
on $p^{\prime}$, then with positive probability $\theta^2$ they will both
charge $p^{\prime}$ and will each have a contribution proportional to $%
(\theta^2/2)(p^{\prime}-0)$ towards their expected profits.  Firm 1 could
increase its expected profit, however, by deviating to putting zero weight
on $p^{\prime}$ and positive weight on $p^{\prime}- \epsilon, $ for
sufficiently small $\epsilon.$ This would replace the expected profit of $%
(\theta^2/2)(p^{\prime}-0)$ with the larger expected profit of $%
(\theta^2)(p^{\prime}-\epsilon )$. Thus, it cannot be that both firms put
positive probability on any $p^{\prime}$ in equilibrium.

Let us then consider a situation in which only Firm 1 chooses $p^{\prime}$
with positive probability mass. There then exists a neighborhood around $%
p^{\prime}$ where prices are not chosen with a strictly positive probability
mass. We distinguish two possibilities. First, there exists a neighborhood $%
[p^{\prime},x)$ with $x>p^{\prime}$ such that the probability that any firm
charges a price in the neighborhood equals 0. This cannot be an equilibrium,
as Firm 1 can increase $p^{\prime}$ without reducing its chance of winning
the customer. Second, there exists a neighborhood $(p^{\prime},x)$ with $%
p^{\prime}<x$ such that the probability that at least one other firm charges
a price in the whole neighborhood is strictly positive. This can also not be
part of an equilibrium, however, as one of the other firms has incentive to
shift probability mass from prices just above $p^{\prime}$ to prices just
below it. Hence, there cannot be any equilibrium in which only one firm puts
strictly positive probability on any single price. In conjunction with the
previous paragraph, this means that there is no equilibrium in which any
firm chooses any price with positive probability mass.

Second, the support over which a firm mixes in equilibrium is connected.
Consider hat would happen if Firm 1 randomized over an unconnected support,
which would include at least two intervals, denoted by $[\beta_1,\gamma_1]$
and $[\beta_2,\gamma_2]$. It is easy to see that an optimal (mixed strategy)
response of Firm 2 does not include prices in the interval $%
[\gamma_1,\beta_2]$. In this case, there exists, however, an $\epsilon>0$
such that Firm 1 will not be indifferent between setting a price of $%
\gamma_1-\epsilon $ and setting a price of $\beta_2+\epsilon $. Thus, a
necessary condition for Firm 1 randomizing over $[\beta_1,\gamma_1]$ and $%
[\beta_2,\gamma_2]$ is violated.

Let us therefore construct an equilibrium in mixed strategies with the
strategies having a continuous and compact support. Let $F(p_i)$ be the
probability that firm $i$ charges a price smaller than $p_i$. The expected
payoff to firm $i$ of charging a price $p_i$ when all other firms choose a
mixed strategy according to $F(p_i)$ is 
\begin{equation}  \label{j1}
\pi_i(p_i,F_i(p)) = \Sigma_{k=0}^{N-1} \left( 
\begin{array}{c}
N -1 \\ 
k
\end{array}
\right)(1-\alpha)^k [\alpha (1-F(p_i)) ] ^{N-k-1} p_i.
\end{equation}

This expression can be explained in the following way. The probability that
exactly $N-k-1$ out of the other $N-1$ firms besides Firm $i$ are active is
equal to 
\begin{equation}  \label{j1.4axx}
\left(
\begin{array}{c}
N -1 \\ 
k
\end{array}
\right) (1-\alpha)^k \alpha^{N-k-1}.
\end{equation}
The expected payoff to firm $i$ when exactly $N-k-1$ firms are active and
when it charges a price of $p_i$ is equal to $p_i$ times the probability
that each of these $N-k-1$ firms charges a price that is larger than $p_i$,
which is $(1-F(p_i))^{N-k-1} p_i$. Multiplying these two terms and summing
up over all $k$ gives the expression above.

Expression (\ref{j1}) is, of course, nothing but an application of the
Binomial Theorem, and a standard result says that 
\begin{equation}  \label{j1.4a}
\Sigma_{k=0}^{N-1 } \left( 
\begin{array}{c}
N -1 \\ 
k
\end{array}
\right) a^k b ^{N-k-1} =(a+b)^{N-1}.
\end{equation}

\noindent  Applying equation (\ref{j1.4a}) to the profit equation (\ref{j1}%
), we obtain 
\begin{equation}  \label{j1.5}
\pi(p_i,F(p_i)) = [1-\alpha F(p_i)]^{N -1} p_i .
\end{equation}

In equilibrium, firm $i$ must be indifferent between all pure strategies
that are in the support of the mixed strategy distribution. Hence, it must
be that on some interval of prices the derivative of expression (\ref{j1.5})
with respect to $p_i$ equals zero. Thus, a necessary condition for any
equilibrium in continuous mixed strategies is 
\begin{equation}  \label{j1.7}
[1- \alpha F(p_i)]^{N-1} - (N -1) [1- \alpha F(p_i)]^{N-2} \alpha f(p_i) p_i
=0,
\end{equation}
or 
\begin{equation}  \label{j2}
1- \alpha F(p_i) - \alpha (N -1)f(p_i) p_i =0,
\end{equation}
where $f$ is the density function associated with cumulative distribution
function $F$.

It is a matter of straightforward calculations to show that the solution to
differential equation (\ref{j2}) is\footnote{%
Note that when there are $m$ identical consumers, the profit in equation (4)
is simply multiplied by $m$ and and the equilibrium price distribution
remains the same.} 
\begin{equation}  \label{j3}
F(p_i) = \frac{ 1 - (1-\alpha) \left( \sqrt[N-1]{\frac{v}{p_i} } \right) }{%
\alpha} ,
\end{equation}
for $(1-\alpha)^{N-1} v \leq p_i \leq v$.

Result (\ref{j3}) implies that there is a unique symmetric equilibrium with
compact support, and we have shown earlier that an equilibrium in pure
strategies does not exist. These results are stated in Proposition 1.

\noindent {\it Proposition 1. The unique symmetric equilibrium of the
Bertrand model with an uncertain number of competitors is in mixed
strategies and the distribution function of a player's strategy is } 
\begin{equation}  \label{j4}
F(p_i) = \left\{ 
\begin{array}{lll}
0 & for & p_i \leq (1-\alpha)^{N-1} v \\ 
&  &  \\ 
\frac{ 1 - (1-\alpha) \left( \sqrt[N-1]{\frac{v}{p_i}} \right) } {\alpha} & 
for & (1-\alpha)^{N-1} v \leq p_i \leq v \\ 
&  &  \\ 
1 & for & p_i \geq v
\end{array}
\right.
\end{equation}

Price dispersion is a well-known outcome in real-world markets. Warner and
Barsky (1995), for example, sampled prices at various stores in Michigan for
a number of identical single products and found considerable dispersion.%
\footnote{%
See Tables I and III of Warner and Barsky (1995). They found, for example,
that a GI Joe had prices of 3.88, 2.93, 2.69, 2.96, 2.84, 2.96, and 2.69,
while and a Huffy Vortex unassembled boy's bicycle had prices of 73.47,
99.99, 112.63, 119.99, 119.99, and 18.70.} Thus, the mixed strategy we found
is quite consistent with reality.

Figure 1 shows the cumulative density for different values of $N$ using
equation (\ref{j4}) with $\alpha =.2$ and $v=100$ (prices are at intervals
of 1, connected). As $N$ increases, each firm chooses relatively low prices
with higher probability. As $N$ becomes large, the cumulative density
function approaches 1 for all values of $p$ that are strictly positive. Of
course, the equilibrium price under perfect competition is also equal to 0.
The perfectly competitive outcome can be regarded as the limit case of the
present model when the number of firms becomes very large.

 \newpage


\begin{center}
{\bf Figure 1: Equilibrium Price Distributions as Industry Concentration
Rises ($\alpha =.2, v =100$) }
\end{center}

The intuition is straightforward. As the number of potential competitors
increases, the probability of at least one other firm actively producing the
same product rises. With greater probability of competition, the firm
reduces its prices. In the limit, a firm is extremely likely to have at
least one active competitor. Standard Bertrand competition comes into effect
and each firm charges a price equal to marginal cost.

Expected profit for one firm can be found using the pure strategy profit
from charging $p=v$. Since the firm is active with probability $\alpha$,
that profit is 
\begin{equation}  \label{a7}
\pi_i = \alpha (1-\alpha)^{N-1} v .
\end{equation}

Note that individual profit is declining in $N$ and its sum, industry
profit, is equal to\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}
N \alpha (1-\alpha)^{N-1} v.
\end{equation}

Let $\Pi_b$ denote expected industry profit under Bertrand competition of
this kind given that at least one firm is active. The profit in equation (%
\ref{a8}) can be written as 
\begin{equation}  \label{a10}
\sum_{i=1}^{N} \pi_i = N \alpha (1-\alpha)^{N-1} v = (1-\alpha)^N (0) + [1-
(1-\alpha)^N] \Pi_b,
\end{equation}
yielding 
\begin{equation}  \label{a11}
\Pi_b= \frac{N \alpha (1-\alpha)^{N-1} v}{1- (1-\alpha)^N}.
\end{equation}

To see how industry profit changes with $N$, note that after some
manipulation, 
\begin{equation}  \label{a13}
\frac{d \Pi_b}{d N} = \left[ \frac{ ( 1-(1-\alpha)^N ) + N log(1-\alpha) } {
( 1-(1-\alpha)^N )^2 }\right] \left[ \alpha (1-\alpha)^{N-1} v \right]
\end{equation}

Derivative (\ref{a13} ) is well-defined, even though only integer values of $%
N$ have an economic interpretation. Its sign is the same as the sign of 
\begin{equation}  \label{a14}
1 - (1-\alpha)^N+ N log(1-\alpha).
\end{equation}

For $N=1$, expression (\ref{a14}) becomes $\alpha + log (1-\alpha)$, which
is negative because $\alpha <1$. For larger $N$, expression (\ref{a14})
becomes even more negative, because its derivative with respect to $N$ is $-
(1-\alpha)^N log (1-\alpha) + log (1-\alpha) = log (1-\alpha) [1-
(1-\alpha)^N] <0$. Thus, 
\begin{equation}  \label{a14a}
\frac{d \Pi_b}{d N} <0,
\end{equation}
and profits fall as the number of firms increases.

In the appendix it is shown that we can say more, namely 
\begin{equation}  \label{a14e}
\frac{d^2 \Pi_b}{d N^2} >0.
\end{equation}

This means that profits are convexly decreasing in the number of firms in
the industry, so the shape shown in the numerical examples graphed in Figure
2 in Section 4 would be found for any example.

\bigskip \noindent {\it More General Demand Structures}

So far the assumption has been that the quantity demanded is one unit for
all prices smaller than $v$ and zero otherwise. Here, we will consider a
more general demand function, which we denote by $D(p)$. For simplicity we
will restrict ourselves to the case $N=2$. We will impose one condition on
this demand function, namely that $pD(p)$ is increasing in $p$ for $p < p_m$
, where $p_m$ is the monopoly price. Most demand function that are commonly
employed satisfy this condition. It is satisfied, for example, if $pD(p)$ is
concave in $p$.

\bigskip

\noindent {\it Assumption 4.1. The function $pD(p)$ is increasing and
differentiable on $[0, p_m)$.}

\bigskip

For general demand functions, the expected profit of firm 1 when firm 2
chooses a price according to the cumulative mixed strategy distribution $%
F_2(p)$ is given by 
\begin{equation}  \label{j5}
\pi_1( p_1, F_2(p_1)) = (1-\alpha)p_1 D(p_1) + \alpha (1-F_2(p_1)) p_1
D(p_1).
\end{equation}

A necessary condition for an equilibrium in mixed strategies with continuous
support to exist is that on a certain domain of prices 
\begin{equation}  \label{j6}
[ (1-\alpha) + \alpha (1-F_2(p_1))] [ D(p_1) + p_1 D^{\prime}(p_1)] -\alpha
f_2(p_1) p_1 D(p_1) = 0.
\end{equation}

\noindent One can show that the solution to differential equation (\ref{j6})
is given by 
\begin{equation}  \label{j7}
F_2(p) = \left\{ 
\begin{array}{lll}
0 & if & p \leq \underline{p} \\ 
&  &  \\ 
\frac{1}{\alpha} \left[ 1- \frac{ (1-\alpha) p_m D(p_m)}{pD(p)} \right] & if
& \underline{p} < p\leq p_m \\ 
&  &  \\ 
1 & if & p > p_m
\end{array}
\right.
\end{equation}

A similar solution holds for Firm 1. It is clear that equation (\ref{j7}) is
similar to equation (\ref{a11}) and the results of the basic model
generalize to more general demand functions. Note that from the solution for 
$F_i(p)$ it is clear why we have to impose a condition on demand: a
necessary and sufficient condition for $F_i(p)$ to be increasing in $p$ is
that $pD(p)$ is increasing in $p$ for all values of $p$ smaller than $p_m$.
In the present case it is impossible to provide an explicit solution for the
domain of prices over which a firm randomizes. It is clear that the upper
bound is given by $p_m$. This is because even if the other firm does not
exist, it is not optimal to set a higher price. The lower bound of the
domain, denoted by $\underline{p}$ , is defined implicitly by the condition $%
\underline{p} D(\underline{p} ) = (1-\alpha) p_m D(p_m)$ As $pD(p)$ is
increasing in $p$ for $p< p_m$, $\underline{p}$ is uniquely defined in this
way.

Industry profits may be calculated as in Section 2 and equal 
\begin{equation}  \label{a15}
\Pi_b= \frac{N \alpha (1-\alpha)^{N-1}p_mD(p_m)}{1- (1-\alpha)^N}.
\end{equation}

\bigskip \noindent {\it 3. Endogenizing Entry}

One way to think about the probability of a firm being active, $\alpha$, is
in a Bayesian way. A given firm and a given customer either make contact or
not, in a way determined by Nature and independent of the number, $N$, of
potential firms. Contact depends on such things as whether a customer
notices the firm's existence in the course of his daily activities (and vice
versa), whether the firm's equipment and labor are working that day, and
whether he knows what potential firms are available. In this case, when
there are $N$ firms, the probability that they all know about the customer
and compete for his business is $\alpha^N$. There are two types of firms:
those that are in active competition for the customer and those that are
not. Those not competing are unimportant; the equilibrium strategy of those
that are indeed actively competing is what we have analyzed in Section 2.

Another way to think about the probability of active competition is that it
stems from a previous decision of the firm (or consumer) which is observable
to other firms setting prices. In the present Section we use this approach,
and consider two ``front-end'' games that endogenize whether a firm is
active. We will concentrate on games with just two potential firms since our
aim is  to illustrate how the probability $\alpha$ in the previous model
might arise, but we will briefly consider the general case of $N$ firms.
Section (i) is a standard model of entry that requires a fixed cost. A firm
does not know whether the other firm has entered when it must choose its
price. Section (ii) is a model of output or capacity. Two firms choose how
much to produce before they set their prices. When setting prices they do
not know the quantity chosen by the other firm. In both models, whether a
given firm is active is random in the symmetric equilibrium.

\bigskip

\noindent {\it (i) A Model with a Fixed Entry Cost}

Consider the following two-stage extension of the basic model. Suppose there
are two potential firms. In the first stage, both firms decide whether or
not they enter the industry. There is a fixed entry cost denoted by $F$ with 
$F$ less than $v$, the consumer's reservation price. At the beginning of the
second stage the firms have not observed whether the other firm has entered
or not. In the second stage, the firms set a price if they entered in the
first stage. One example is a sealed bid auction with an entry fee, a common
situation in government procurement: it is costly to prepare a bid, and when
sending in their bids firms do not know how many competing bidders there
are. As the outcome of the first stage is not observed, we can analyze the
game as a simultaneous move game.

There are three equilibria. In the two asymmetric equilibria, one firm
enters the market and sets a price equal to $v$, while the other firm stays
out. In the third, symmetric, equilibrium, both firms are indifferent
between entering the market or staying out and they enter the market with a
certain probability $\gamma$. Given this probability of entering, each firm
chooses a price according to the mixed strategy distribution calculated in
Section 2, with $\gamma$ replacing $\alpha$. The expected payoff in the
second stage is $(1-\gamma)v$. The only way in which the firms can be
indifferent between staying out and entering the market is if $(1-\gamma)v$
equals the fixed entry cost $F$. Thus, $\gamma$ equals $1-F/v$, and expected
profits are zero.

This model can easily be extended to $N$ potential firms. Then, as with two
firms, there will be a symmetric mixed-strategy equilibrium and a number of
asymmetric equilibria. The only novelty is that mixed-strategy asymmetric
equilibria can exist if there are more than 2 potential entrants; with 3
firms, for example, in equilibrium one firm might enter with probability 1
and the other two firms would mix. For general values of $N$, the endogenous
parameter $\gamma$, does, however, ordinarily depend on market size. If we
increase $N$ while maintaining the assumption that there is only one
consumer, it is easy to see that the $\gamma$ will decrease. A more
reasonable comparison, however, is to increase the market size at the same
time as $N$, in which case what happens depends on how fast $N$ increases
with market size. If we denote $m_N$ as the size of demand when there are $N$
firms, then $\gamma$ does not depend on market size only if $m_{N+1} = \frac{%
1}{1-\gamma} m_N$, a special case. This should be kept in mind while reading
our discussion of empirical profit-concentration relationships below,
because there as $N$ increases we keep constant the probability a given firm
is active, rather than either increasing or decreasing it.

\bigskip \noindent {\it (ii) A Model of Output Choice}

Kreps and Scheinkman (1983) describe a model in which firms compete first in
outputs and then in prices, something we can do here also.  Consider a
market with two consumers, each buying up to one unit each and with a
reservation price of $v$. There are two firms, each of whom can decide in
the first stage whether to produce 1 or 2 units of a homogeneous output. In
the second stage, firms compete in prices not knowing the decision of the
other firm in the first stage. The cost of producing 1 unit is normalized to
0 and the cost of producing 2 units is $K$, where $0<K<v$.

There does not exist an equilibrium in pure strategies. It is not an
equilibrium for Firm 1 to produce 1 unit and charge a high price, because
Firm 2 would respond with 2 units and a slightly lower price, driving Firm
1's profits to zero. There cannot be one in which Firm 1 chooses to produce
1 unit and charge a low price, because Firm 2 would respond with 1 unit and
a price of $v$, making it profitable for Firm 1 to deviate and produce 2
units and charge slightly less than $v$. Nor can it be an equilibrium for
Firm 1 to produce 2 units and charge a price greater than $K$, since Firm 2
would produce 2 units also and charge a lower price, making Firm 1
unprofitable. Finally, it cannot be an equilibrium for Firm 1 to produce 2
units and charge a price of $K$ or less, since Firm 2's best response would
be to produce 1 unit and undercut Firm 1's price, in which case Firm 1 would
do better to produce 1 unit and charge a price of $v$.

Consider an equilibrium in mixed strategies: Each firm chooses with
probability $\mu $ to produce 2 units, and otherwise produces 1 unit. If a
firm happens to choose 1 unit, it will charge a price equal to $v$. The
profit of any firm producing 2 units is then 
\begin{equation}  \label{z1}
\pi_i (p_i^2, F^1_{-i} (p_i) , F^2_{-i} (p_i^2) ) = \left\{ 
\begin{array}{lll}
2 (1-\mu) p_i^2 + 2 \mu (1-F^2_{-i} (p_i)) p_i^2 - K & if & p_i^2 <v \\ 
&  &  \\ 
\frac{4}{3} (1-\mu) v + 2 \mu (1-F^2_{-i} (p_i^2))v - K & if & p_i^2 =v
\end{array}
\right.
\end{equation}
where $p_i^2$ is the price charged by firm $i$ if it has produced 2 units
and $F^j_{-i} (p_i^2)$ is the distribution function with which Firm $-i$
chooses prices if it has produced $j$ units.

It is easy to see that it cannot be optimal to set $p_i^2=v$. A necessary
condition for an equilibrium in mixed strategies is 
\begin{equation}  \label{z2}
(1-\mu) + \mu (1- F^2_{-i} (p_i^2))- \mu f^2_{-i} (p_i^2)p_i =0,
\end{equation}
or 
\begin{equation}  \label{z3}
1-\mu F^2_{-i} (p_i^2 ) - \mu f^2_{-i} (p_i^2)p_i =0,
\end{equation}
This equation has the same form as equation (5) for $N=2$. Hence, the
solution is given by 
\begin{equation}  \label{z4}
F^2_{-i} (p_i^2)= \frac{ 1 - (1-\mu) \frac{v}{p_i^2} }{\mu}
\end{equation}
for $(1-\mu) v < p_i < v$.

The profit to either firm of producing 1 unit is 
\begin{equation}  \label{z5}
\pi_i (p_i^1, F^1_{-i} (p) , F^2_{-i} (p) )= [ 1-\mu + \mu (1- F^2_{-i}
(p^1_i)] p^1_i,
\end{equation}
where $p^1_i$ is the price charged by firm $i$ if it has produced 1 unit.

It is easy to see that given equation (\ref{z4}) the right-hand-side of
equation (\ref{z5}) reduces to $(1-\mu)v$, which is independent of $p^1_i$.
Hence, given (\ref{z4}), it is optimal to set $p^1_i=v$.

The profits of producing 1 and 2 units are $(1-\mu)v$ and $2(1-\mu)v-K$. In
the mixed strategy equilibrium, these two expressions have to be equal to
each other, which implies that $\mu=(v-K)/v$.

So, the entry decision-- here, with how many units to enter the Bertrand
competition phase-- is random, and the uncertainty about the existence of a
fierce competitor has been derived endogeneously.\footnote{%
Note that there exists a continuum of equilibria,all in mixed strategies of
the form derived in Section 2, indexed by the price a firm sets when
producing only 1 unit.}

\noindent {\it 4. Comparing Bertrand and Cournot}

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. Even though the assumptions of price competition
seem more realistic, the quantity model yields more realistic outcomes,
because profits are positive, but fall gradually as the number of firms
increases. We have shown that this objectionable feature of the Bertrand
model disappears when uncertainty about the presence of competitors is taken
into account. We will now see what happens to the Cournot model when
uncertainty is added, and compare the Bertrand and Cournot models under
uncertainty. To make the comparison clearer, we will use linear demand, 
\begin{equation}  \label{b1}
p\left(\sum_{i=1}^{N} q_{i}\right) = a - b \sum_{i=1}^{N} q_{i}.
\end{equation}

\noindent  Let us define $q(p)$ as the demand facing a monopolist at a price
of $p$, so 
\begin{equation}  \label{b2}
q(p) = \frac{ a}{b} - \frac{ p}{b}.
\end{equation}

The monopoly price then equals $a/2$ and the quantity demanded is $a/2b $ at
that price.

We will compute the expected profits from Cournot and Bertrand for different
levels of $N$ to obtain some idea of the effects of concentration in each.

\newpage \vspace*{36pt}

\bigskip

\noindent {\it Bertrand equilibrium}

Applying equation (\ref{a15}) to the case of linear demand, the industry
profits in the Bertrand model with uncertainty are

\begin{equation}  \label{b7}
\Pi_{bertrand}= \frac{ N \alpha (1-\alpha)^{N-1} p_m D(p_m)}{ 1-
(1-\alpha)^{N} } =\frac{ N \alpha (1-\alpha)^{N-1} \frac{a^2}{4b}}{ 1-
(1-\alpha)^{N} }.
\end{equation}

Adding uncertainty eliminates the discontinuous behavior of the original
Bertrand model. Uncertainty makes a big difference, and the comparative
statics become consistent and intuitive. Profits are always positive, but
they fall whenever the number of firms or the probability of more firms
being active increases. Figure 2 shows this for a particular numerical
example with $a=100$, $b =1$, $N$ from 0 to 7, and $\alpha $ from 0 to 1.%
\footnote{%
In every case, expected industry profits are conditional upon at least one
firm being active. When $\alpha =0 $, this is to be interpreted as the
probability zero (but nonetheless possible) event that one firm is active
and the expected number of other firms is zero.}

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\begin{center}
{\bf Figure 2: Bertrand Profits For Different Probabilities of Activity, $%
\alpha$, and Numbers of Firms, $N$} \\[0pt]
(from Equation (\ref{b7}  (conditional on at least one firm being active)
\end{center}

\vspace*{36pt}

\bigskip

\noindent {\it Cournot Equilibrium}

The Cournot equilibrium is calculated the same way as in the standard linear
Cournot model except that we must account for the possibility that the
number of active firms might be anywhere from 1 to $N$. Let $q^*$ be the
Cournot output we are trying to determine. Firm $i$'s expected profit if all
other firms choose $q^*$ is the sum of his profits for each possible number
of active firms times the probability exactly that many firms are active,

\begin{equation}  \label{b8}
\begin{array}{ll}
\pi_i(q_i,q^*) & = \sum_{j=0}^{N-1} \left( 
\begin{array}{c}
N -1 \\ 
j
\end{array}
\right) (1-\alpha)^j \alpha^{N-1-j} [p(q_i + (N-1-j)q^*) ]q_i.
\end{array}
\end{equation}

Substituting in the linear demand function, differentiating with respect to $%
q_i$, setting $q^*=q_i$, and solving for $q^*$ yields the equilibrium
expected Cournot industry profit conditional upon one firm being active,%
\footnote{%
Equation (\ref{b8c}) is conditional upon $Nq^*$ being not 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 $N$
firms coincidentally being active and producing a large amount.} 
\begin{equation}  \label{b8c}
\Pi_{Cournot} = \frac{a^2 \alpha N }{ b[1-(1-\alpha)^N] [2+ \alpha (N-1)]^2}.
\end{equation}

Figure 3 depicts Cournot profits for different degrees of activity and
concentration, using the same numerical parameters as the Bertrand profits
in Figure 2.

\bigskip 

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\begin{center}
{\bf Figure 3: Cournot Profits For Different Probabilities of Activity, $%
\alpha$, and Numbers of Firms, $N$ }\\[0pt]
(from Equation (\ref{b8c}),  conditional on at least one firm being active)
\end{center}

\vspace*{36pt}

Figure 3 shows that depending on the number of firms in the industry, the
presence of uncertainty to the Cournot model can either increase or reduce
industry profits, but it does not radically change the equilibrium. Under
Cournot competition, a firm expands its output when it expects fewer rivals
to be helping push down the price and the next effect on expected industry
output is unclear. Conflicting forces are at work in Cournot equilibrium,
and the net result is sensitive to particular values of the parameters
underlying the model.\footnote{%
The result is reminiscient of the peculiarities of profit per firm in the
Cournot model, which can (but do not always) give rise to an incentive for a
Cournot firm to split in two to increase its profits, as noted by Salant,
Switzer and Reynolds (1983).}

\bigskip

\noindent {\it Profits and Concentration in Bertrand, Cournot, and the
Bresnahan-Reiss Study}

Let us now compare Bertrand and Cournot. Using profit equations (\ref{b7})
and (\ref{b8c}), the ratio of industry profits under Bertrand and Cournot
competition is

\begin{equation}  \label{b8d}
\frac{\Pi_{Bertrand}}{\Pi_{Cournot}} = (1-\alpha)^{N-1} \left[1 + \frac{%
\alpha}{2} \left( N-1 \right) \right]^2,
\end{equation}

which is decreasing in both $N$ and $\alpha$.

Table 1 and Figure 4 show the outcomes of our numerical example for
different degrees of concentration under Cournot and Bertrand behavior with
certainty and with $\alpha =.8$. (Figure 4 also illustrates the
Bresnahan-Reiss empirical result, of which more will be said later.) As we
have seen, uncertainty changes the Bertrand model in a 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 in the
Bertrand model is perhaps not so unreasonable as it looks. It is extreme,
but it is a limiting result as $\alpha $ goes to one, as Figures 2 and 4
illustrate.

\begin{tabular}{l|rrrrrrr}
\hline\hline
Number of Firms $N$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline
&  &  &  &  &  &  &  \\ 
Bertrand, $\alpha =1$ & 2500 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
Bertrand, $\alpha =.8$ (eq. (\ref{b7})) & 2500 & 833 & 242 & 64 & 16 & 4 & 1
\\ 
&  &  &  &  &  &  &  \\ 
Cournot, $\alpha =1$ & 2500 & 2222 & 1875 & 1600 & 1388 & 1224 & 1093 \\ 
Cournot, $\alpha =.8$ (eq. (\ref{b8c})) & 2500 & 2125 & 1867 & 1650 & 1480 & 
1333 & 1211 \\ 
&  &  &  &  &  &  &  \\ \hline\hline
\end{tabular}

{\bf Table 1: Industry Profits for Different Concentration Levels}\footnote{%
Numerical calculations and figures use {\it Mathematica}. Values are rounded.%
}

\bigskip

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\begin{center}
{\bf Figure 4: Bertrand and Cournot Profits}
\end{center}

Consider the shape of the profit-concentration paths. All the curves in
Figures 2 through 4 have convex shapes, if only weakly in the limiting
cases, but the curvatures, and therefore the empirical implications, are
different. As Figure 4 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 2500 to duopoly profits of
833, triopoly profits of 242, and low levels thereafter. Cournot profits
show a much more uniform decline as concentration falls.

Comparison of Figures 2 and 3 shows that for smaller values of the  activity
probability $\alpha $ the Bertrand profit path becomes flatter and the
Cournot path, perhaps 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{%
See pp. 349-366 of Carlton and Perloff 's 1994 industrial organization text
for a good discussion of the problems of the profits-concentration
literature.} 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.

Bresnahan and Reiss (1991) took a clever empirical approach to the same
problem. They took the unit of observation to be the market for a particular
product 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.

\begin{tabular}{l|rrrrr}
\hline\hline
Number of Firms $N$ & 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 }

{\bf (1,000's of inhabitants)}\footnote{%
Calculated from Table 5A of Bresnahan  and 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

Bresnahan and Reiss used this approach to estimate the thresholds $s_i$ 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.
\par
Define the monopoly level of profits in an industry to be 2500, 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.

\begin{tabular}{l|rrrrr}
\hline\hline
Number of Firms, $N$ & 1 & 2 & 3 & 4 & 5 \\ \hline
&  &  &  &  &  \\ 
Doctors & 2500 & 430 & 0 & 0 & 0 \\ 
Dentists & 2500 & 440 & 0 & 0 & 0 \\ 
Druggists & 2500 & 1550 & 430 & 0 & 0 \\ 
Plumbers & 2500 & 830 & 830 & 0 & 0 \\ 
Tire Dealers & 2500 & 1130 & 270 & 100 & 0 \\ 
&  &  &  &  &  \\ \hline
Average & 2500 & 960 & 230 & 20 & 0 \\ \hline\hline
\end{tabular}

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

What is significant is how profits flatten out, even though the choice of 0
as the flat level in Table 3 is arbitrary. The empirical result that
full-fledged competition kicks in  quickly, so going from one firm to two is
much more important than going from two to three, matches the Bertrand model
with uncertainty very well but is inconsistent with the Cournot model. And
this is true even though we have maintained our assumption that the
probability a firm is active is constant as $N$ varies. If we took that
probability to be a free variable, we could of course fit the data even
better. As it is, we see a correspondence between the Bresnahan-Reiss
findings and one version of our model: the simple version in which the
probability a given firm is active is exogenous and hence neither rises nor
falls with the size of the market.

We do not want to argue that the Bertrand model with uncertainty is the only
model that may explain the data presented. Other models in the industrial
organization literature may also explain the data. By no means all models
do, however. For example, consider the model of  Salop (1979) as set out in
Tirole (1988, pp.282-4) with linear and/or quadratic cost for an exogenously
determined value of $N$. With linear (quadratic) cost, the relation between
price and $N$ is given by $p=c+t/N$, respectively $p=c+t/N^2$, where $t$ is
a transportation cost parameter and $c$ is marginal cost. As toptal demand
is constant in the Salop model, industry profits are given by $t/N$, and $%
t/N^2$, respectively Confronting these equations with the Bresnahan and
Reiss results reveals that profits in the Salop model do not decline
exponentially, as our model and the empirical results suggests.

\bigskip

\noindent  {\it 5. Concluding Remarks}

The Bertrand model with uncertainty about the number of competitors is
simple, but its properties are both interesting and useful, and, in
particular, the extreme transition from monopoly to competition found in the
standard Bertrand model disappears. Expected profits are positive, but
decline with the number of firms in the industry, and decline in a way that
empirical work suggest is more realistic than the way they decline in the
Cournot model. We have tried to show that the model is useful both as a
simple description of oligopoly and as a building block for other topics in
industrial organization, and this usefulness has already been illustrated in
Gwin (1997), and Janssen and Van Reeven (1998).

\newpage

\noindent {\it Appendix on Convexity}

This appendix shows that Bertrand industry profits are convex in $N$.

The second derivative $\frac{d^2 \Pi_b}{d N^2}$ is derived from the first
derivative in (\ref{a13}), which can be rewritten as 
\begin{equation}  \label{a115}
\frac{d \Pi_b}{d N} = \alpha v \left\{ \frac{(1-\alpha)^{N-1}} {
1-(1-\alpha)^N } + \frac{ (1-\alpha)^{N-1} N log (1-\alpha)} {
[1-(1-\alpha)^N]^2 } \right\}.
\end{equation}

\noindent  The derivative of this is 
\begin{equation}  \label{a116}
\begin{array}{ll}
\frac{d^2 \Pi_b}{d N^2} & = \alpha v \left\{ \frac{ [1-(1-\alpha)^N
](1-\alpha)^{N-1}log (1-\alpha)+(1-\alpha)^N (1-\alpha)^{N-1}log (1-\alpha)%
} { [1-(1-\alpha)^N]^2 } + \right. \\ 
&  \\ 
& \;\;\;\;\;\;\;\; \left. \frac{ [(1-\alpha)^{N-1}log (1-\alpha) +
(1-\alpha)^{N-1} N log^2 (1-\alpha) ] [ 1-(1-\alpha)^N ]^2 + 2 [
1-(1-\alpha)^N ] (1-\alpha)^N log (1-\alpha) [N (1-\alpha)^{N-1} log
(1-\alpha)] } { [1-(1-\alpha)^N]^4 } \right\}. \\ 
&  \\ 
& = \alpha v \left\{ 2\frac{ (1-\alpha)^{N-1}log (1-\alpha)}{
[1-(1-\alpha)^N]^2 }+ \frac{ (1-\alpha)^{N-1} N log^2 (1-\alpha) [ 1-
2(1-\alpha)^N ] + (1-\alpha)^{2N } + 2 (1-\alpha)^N -2(1-\alpha)^{2N } } {
[1-(1-\alpha)^N]^4 } \right\} \\ 
&  \\ 
& = \alpha v \left\{2 \frac{ (1-\alpha)^{N-1}log (1-\alpha)}{
[1-(1-\alpha)^N]^2 }+ \frac{ (1-\alpha)^{N-1} N log^2 (1-\alpha) [ 1-
(1-\alpha)^{ 2N} ] } { [1-(1-\alpha)^N]^4 } \right\} \\ 
&  \\ 
& = \frac{ (1-\alpha)^{N-1}log (1-\alpha)}{ [1-(1-\alpha)^N]^2 } \left\{ 2+%
\frac{ N log (1-\alpha) [1+(1-\alpha)^{N}]}{1 - (1-\alpha)^{N}} \right\}
\alpha v .
\end{array}
\end{equation}

\noindent  The first term of this expression is negative because $log
(1-\alpha)$ is negative.

\noindent  The second term has the same sign as 
\begin{equation}  \label{a118}
2- 2(1-\alpha)^{N} + N log (1-\alpha) [1+(1-\alpha)^{N} ].
\end{equation}

We will show that expression (\ref{a118}) is also negative for all $N$ and
all $\alpha \in (0,1)$. We will first show that it is negative for $N=1$. In
this case we can define $f( \alpha)= 2 \alpha + (2 - \alpha) log(1-\alpha)$.
It is easy to see that $f(0) = f^{\prime}(0) = 0$ and that $%
f^{\prime\prime}(\alpha) = -\frac{\alpha}{ (1-\alpha)^2}$, which is strictly
negative for all $\alpha >0$. Hence, for all $\alpha \in (0,1)$, $f(\alpha)
<0$.

Let us then consider for fixed $\alpha$, 
\begin{equation}  \label{w1 }
g(N) = 2 - 2(1-\alpha)^N + N log (1-\alpha) [1 + (1-\alpha)^N].
\end{equation}
It can be shown that $g^{\prime}(N)$ has the sign of 
\begin{equation}  \label{w1 }
(1-\alpha)^N -1 - (1-\alpha)^N N log (1-\alpha)
\end{equation}
and that $g^{\prime\prime}(N)$ has the sign of 
\begin{equation}  \label{w1 }
-N (1-\alpha)^N log^2 (1-\alpha).
\end{equation}
As $g(1) $,$g^{\prime}(1)$, and $g^{\prime\prime}(N)$ are strictly negative,
we can conclude that expression (\ref{a118}) is negative, so that

\begin{equation}  \label{a120}
\frac{d^2 \Pi_b}{d N^2} >0.
\end{equation}

\newpage \noindent {\it Appendix on Comparison of Bertrand and Cournot
profits}

This appendix shows that the ratio (\ref{b8d}) is decreasing in $N$ and $%
\alpha$. To see the first, take the derivative with respect to $N$, which is 
\begin{equation}  \label{b8e}
\begin{array}{l}
log (1-\alpha)(1-\alpha)^{N-1} \left[ 1 + \frac{\alpha}{2} \left( N-1
\right) \right]^2 +\alpha (1-\alpha)^{N-1} \left[ [1 + \frac{\alpha}{2}
\left( N-1 \right) \right] \\ 
= \{ log (1-\alpha) \left[ [1 + \frac{\alpha}{2} (N-1) \right] +\alpha \}
(1-\alpha)^{N-1} \left[ [1 + \frac{\alpha}{2} \left( N-1 \right) \right]
\end{array}
\end{equation}

The sign of is derivative (\ref{b8e}) is determined by the sign of the first
term. Since 
\begin{equation}  \label{b8f}
\left[ 1 + \frac{\alpha}{2} \left( N-1 \right) \right] \geq 1 > \frac{ -
\alpha}{log (1-\alpha)},
\end{equation}
the derivative is negative.

To see that ratio (\ref{b8d}) is decreasing in $\alpha$, take the derivative
with respect to $\alpha$, which is 
\begin{equation}  \label{b8g}
\begin{array}{l}
-(N-1) (1-\alpha)^{N-2} \left[ 1 + \frac{\alpha}{2} \left( N-1 \right) 
\right]^2 + (N-1) (1-\alpha)^{N-1} \left[ 1 + \frac{\alpha}{2} \left( N-1
\right) \right] \\ 
= -(N-1) (1-\alpha)^{N-2} \left[ 1 + \frac{\alpha}{2} \left( N-1 \right) 
\right] \left[ 1 + \frac{\alpha}{2} \left( N-1 \right) - (1-\alpha) \right]
\\ 
= -(N-1) (1-\alpha)^{N-2} \left[ 1 + \frac{\alpha}{2} \left( N-1 \right) 
\right] \left[ \frac{\alpha}{2} \left( N+1 \right) \right]
\end{array}
\end{equation}
which is negative.

\newpage

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