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{\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 $%
p^{\prime}0$
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 $00$. 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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