Bertrand, Cournot and Monopolistically Competitive Equilibria
Richard Kihlstrom
Revised, August, 1999
Abstract. Using an example, we study the analogs, for the di¤erentiated
product case, of the Cournot and Bertrand equilibria. These equilibria can be
shown to exist and be unique if we impose a simple and natural restriction on
the elasticities of the demand functions for the di¤erentiated products. Our
characterizations of these equilibria make it possible to compare them and to
determine how they are a¤ected by the size of the market and the number of
…rms. We are also able to prove the existence of Cournot free-entry equilibria
in which the number of …rms is determined endogenously. In addition, we
are able to prove that, in a large market, the Cournot free-entry equilibria
approximate the Dixit-Stiglitz monopolistically competitive equilibria. The
free-entry equilibrium concept we study is an analog of the one studied by
Novshek for the case of …rms selling products that are perfect substitutes. Our
results are extensions of Novshek’s. While we were unable to establish a general
existence result for Bertrand free-entry equilibria, we were able to prove that,
when these equilibria exist, they are unique and that in large markets they also
approximate the Dixit-Stiglitz equilibria.
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outline of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to the Model . . . . . . . . . . . . . . . . . . . . . . . . .
The Chamberlin-Dixit-Stiglitz Equilibrium . . . . . . . . . . . . . . . .
Cournot Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cournot-Free Entry Equilibrium . . . . . . . . . . . . . . . . . . . . . .
Bertrand Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Collusive Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bertrand and Cournot Equilibria and the Collusive Outcome Compared
Bertrand-Free Entry Equilibrium . . . . . . . . . . . . . . . . . . . . .
A Case When Our Elasticity Condition Fails . . . . . . . . . . . . . . .
Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bertrand, Cournot and Monopolistically Competitive Equilibria
2
1. Introduction
The aim of this paper is to study a market for di¤erentiated products. In the model
employed, the demands for the di¤erentiated goods are derived using the same approach as that taken by Dixit and Stiglitz in their analysis [2] of Chamberlin’s monopolistically competitive equilibrium [1]. Our purpose, however, is to study these
markets when the number of …rms is …nite and may be small. Thus, we are interested
not only in the Chamberlinian equilibrium, but also in the analogs, for the di¤erentiated product case, of the Cournot and Bertrand equilibria. These equilibria can be
shown to exist and be unique if we impose a simple and natural restriction on the elasticities of the demand functions for the di¤erentiated products. Under this demand
function restriction, it is also easy to compare the Bertrand and Cournot equilibria
and to show that the Bertrand prices and pro…ts are lower and the Bertrand supplies are higher than those arising in the Cournot equilibria. The demand restriction
imposed to obtain these results is simply that the elasticity of demand in the DixitStiglitz monopolistically competitive equilibrium is higher than in the monopoly case
in which only one …rm produces. This turns out to be a condition that Chamberlin
and Dixit-Stiglitz also imposed. In particular, Dixit and Stiglitz refer to ”Chamberlin’s DD and dd curves.” The dd curve is the Dixit-Stiglitz demand curve faced
by each …rm in the monopolistically competitive equilibrium. The DD curve is the
demand curve each …rm would face if the …rms colluded to charge the same price. Following Chamberlin, Dixit and Stiglitz assume that the dd curve is more elastic than
the DD curve, a condition that is easily seen to be identical to the one we impose.
For the purpose of comparing the per …rm pro…ts obtained in the Bertrand and
Cournot equilibria it is convenient to compare them both to the collusive outcome
that arises when …rms choose a common price and maximize per …rm pro…ts which
are computed using the fact that, at any common price, each …rm’s sales will be
given by the DD curve. As expected, the collusive prices and pro…ts are higher and
the collusive output levels are lower than those arising in the Cournot and Bertrand
equilibria.
We also study Cournot and Bertrand analogs of Novshek’s version [3] of the
Cournot equilibrium in which the number of …rms is determined by free entry. Novshek
studied the case in which a typical potential …rm produced with a technology that
gave rise to a ”U-shaped” average cost curve. While simply assuming that the demand curve sloped down, he was able to prove that if a typical potential …rm’s average
cost minimizing output level is ”small”relative to market demand when price equals
minimum average cost, then the total amount supplied in the free entry Cournot
equilibrium is approximately the ”competitive output” and the price is, therefore,
approximately the ”competitive price;”i.e., the price is approximately equal to minimum average cost. Novshek’s result demonstrated that even if …rms don’t take prices
Bertrand, Cournot and Monopolistically Competitive Equilibria
3
as given, entry will drive the noncompetitive Cournot equilibrium price to the competitive price if the typical …rm’s e¢ cient scale is small relative to the size of the
market.
Like the competitive equilibrium, the monopolistically competitive equilibrium,
is commonly viewed as an approximation to a noncompetitive equilibrium in which
entry drives price down to average cost. Dixit and Stiglitz also argue that, in their
formalization of the monopolistically competitive model, the presence of a large number of …rms drives the elasticity of demand to a …nite limiting value that they derive.
Our aim is follow Novshek’s approach and to provide a formalization of a natural
noncompetitive equilibrium in which entry does, indeed, drive price to average cost
and the demand elasticity to its Dixit-Stiglitz limit. For this purpose we de…ne, for
the di¤erentiated products case, Cournot and Bertrand analog’s of Novshek’s freeentry equilibrium. Under the above mentioned restriction on the demand elasticity,
we are then able to establish analogs of Novshek’s result for the case of monopolistic
competition. In particular, we can demonstrate that, if the market demand is large
and if the Cournot and Bertrand free-entry equilibria exist, then in both of these
equilibria, there will be a large number of …rms producing di¤erentiated products
at approximately the Chamberlinian output levels and supplying them at approximately the Chamberlinian price. We are also able to demonstrate that, if the market
is large enough to guarantee pro…ts when only two …rms produce at their Cournot
duopoly output levels, then there will always exist a Cournot free-entry equilibrium.
We are not able to demonstrate that this equilibrium is unique, however. We are
also unable to demonstrate that a Bertrand free-entry equilibrium always exists even
if the market is large. It seems that for some market sizes a Bertrand free-entry
equilibrium will exist and for some it will not. We do demonstrate, however, that
if a Bertrand free-entry equilibrium does exist, it is unique. We can also show that
when a Bertrand free-entry equilibrium exists, there is always a Cournot free-entry
equilibrium in which the number of …rms producing is at least as large as in the
Bertrand free-entry equilibrium.
While we do not extensively analyze the Cournot and Bertrand equilibria for
the case in which our demand elasticity condition fails, we do show that, in the
borderline case in which the Dixit-Stiglitz elasticity equals the monopoly elasticity,
entry has no e¤ect on …rm pro…ts. In that case, pro…ts may or may not be positive
depending on the size of the market. If pro…ts are positive they remain positive
even if there are many …rms. As a consequence, the Dixit-Stiglitz monopolistically
competitive equilibrium cannot be approximated by either a Cournot or Bertrand
free entry equilibria because these equilibria must fail to exist. In this border line
case, there is no reason to expect market forces, in particular, the possibility of entry,
to lead to equilibria that approximate monopolistic competition.
Finally, it should be mentioned that our analysis is restricted to an example. It
Bertrand, Cournot and Monopolistically Competitive Equilibria
4
is speci…cally the case in which the monopoly elasticity of demand is constant. The
constancy of the monopoly elasticity of demand complements the constancy the DixitStiglitz elasticity of demand to yield not only a tractable example but also a canonical
one. Because the example is canonical in this sense, there is hope for supposing that
the arguments developed in the analysis of this example can be extended. In the
summary, we brie‡y discuss the possibility of obtaining such extensions.
2. Outline of the Paper
The demand functions and the inverse demand functions faced by the sellers of differentiated products in the Bertrand and Cournot equilibria are described in Section
2.1. The inverse demand functions are introduced in the subsection on ”The Cournot
Case”and the demand functions in the subsection on ”The Bertrand Case.” A series
of remarks describing some features of the demand and inverse demand functions follows their introductions. These remarks also relate the demand and inverse demand
functions to the Chamberlinian dd and DD curves mentioned in the introduction. In
addition, the remarks form the basis for the discussion in the subsection that follows
in which the parameters are interpreted and the elasticity restriction mentioned in
the introduction is discussed formally. Next is a subsection in which we discuss the
derivation of the demand and inverse demand functions from utility maximization.
The …nal subsection of section 2.1 derives expressions for the elasticities of demand
from both the demand and inverse demand functions. These expressions are discussed
and interpreted in another series of remarks. The cost assumptions are described in
Section 2.2.
The Dixit-Stiglitz equilibrium is de…ned and characterized in Section 3. Since
we make quite speci…c assumptions about demand we are able to go farther than
Dixit and Stiglitz were in describing their equilibrium. In particular we are able to
determine the equilibrium number of …rms and relate the number of …rms to the size
of the market. That relationship is discussed in Section 3.2. Section 3.3 presents two
remarks that contain a comparative static analysis describing the relationship between
the Dixit-Stiglitz equilibrium and the Dixit Stiglitz demand elasticity. Section 3.3
represents a detour from a our main purpose but is included since the relationship
between the equilibrium number of …rms and the Dixit Stiglitz demand elasticity is
somewhat surprising.
For a …xed number of …rms, the Cournot equilibrium is de…ned and characterized
in the beginning of Section 4. The symmetric Cournot equilibrium is described in
Proposition 1 of Section 4.1. In Proposition 2 of Section 4.2, the symmetric equilibrium is shown to be the unique Cournot equilibrium. Proposition 3 of Section 4.3
presents comparative static results relating the Cournot equilibrium price and output
levels to the size of the market and the number of …rms. Corollary 4 to Proposition
3, which begins Section 4.4, describes the e¤ect of market size and the number of
Bertrand, Cournot and Monopolistically Competitive Equilibria
5
…rms on the Cournot pro…ts. Proposition 5 demonstrates that, for any number of
…rms, if the market is large enough, the Cournot pro…ts will be positive. Proposition
6, which concludes Section 4.4 shows that, for any market size, Cournot pro…ts will
be negative if there are many …rms.
A Cournot free-entry equilibrium is de…ned at the beginning of Section 5, and
Proposition 8 demonstrates that for virtually any market size, such an equilibrium
exists. Proposition 9 demonstrates, that when the market is large the Cournot freeentry equilibrium approximates the Dixit-Stiglitz equilibrium.
The Bertrand equilibrium is de…ned, for a …xed number of …rms, in the beginning
of Section 6. A general characterization follows in Remark 39. Proposition 10 of
Section 6.1 describes the symmetric Bertrand equilibrium which is shown to be the
unique Bertrand equilibrium in Proposition 11 of Section 6.2.
Proposition 12 of Section 6.3 is the Bertrand analog of Proposition 3. It presents
the comparative static results relating the Bertrand price and output levels to the
size of the market and the number of …rms. Corollary 13 to Proposition 12 which
begins Section 6.4 shows that Bertrand pro…ts increase with the size of the market.
Proposition 15 demonstrates how entry and the number of …rms a¤ect the Bertrand
pro…ts. Proposition 16 is the Bertrand analog of Proposition 5. It demonstrates that,
for any number of …rms, if the market is large enough, the Bertrand pro…ts will be
positive.
The discussion of the Bertrand free-entry equilibrium, is facilitated by a comparison of the Cournot and Bertrand equilibria, and in making this comparison it
is useful to compare each of these equilibria with the collusive outcome discussed in
the introduction. The collusive outcome is de…ned in Section 7 and characterized in
Proposition 17 of that section. Proposition 18 of Section 8 presents the comparison
between the collusive outcome and the Cournot and Bertrand equilibria. For the
discussion of the Bertrand free-entry equilibrium, the most important result is that
the Bertrand pro…ts are lower than the Cournot pro…ts. This gives us Corollary 19 to
Propositions 6 and 18 which asserts that, for any market size, Bertrand pro…ts will
negative if there are many …rms.
A Bertrand free-entry equilibrium is de…ned at the beginning of Section 9. Although we are unable to show that such an equilibrium exists we are able to demonstrate that, if it exists, it is unique. This is done in Proposition 20. Proposition 21
demonstrates, that when the market is large and a Bertrand free-entry equilibrium
exists it approximates the Dixit-Stiglitz equilibrium.
Section 10 discusses a case in which our equilibrium condition fails and shows that,
in that case, we will typically fail to have existence of either a Cournot or Bertrand
free-entry equilibrium in which …rms produce.
Bertrand, Cournot and Monopolistically Competitive Equilibria
3.
3.1.
6
Introduction to the Model
Demand.
The Cournot Case. Analysis of the Cournot Equilibrium begins with a description of the inverse demand function faced by each …rm. We assume that when
there are n …rms, the inverse demand function faced by …rm s is
!( 1 1 )[ (1 )]
n
X
(1)
ps (x1 ; : : : ; xn ) = mxs
x1t
t=1
n
X
= mxs
!( 1 )
x1t
t=1
where
1
xt = the supply of …rm t;
ps = the price the market will pay for good s;
m>0
and
0<
<1
(2)
< 1:
Remark 1. It is straightforward to verify that
@ps (x1 ; : : : ; xn )
=
@xs
ps (x1 ; : : : ; xn )
xs
This expression is negative if both
(2) fails.
However, when t 6= s;
@ps (x1 ; : : : ; xn )
= m[
@xt
1
x1
Pn s
t=1
and 1
(1
x1t
+ (1
)
x1
Pn s
t=1
x1t
:
are positive, and this is true even if
)] xt xs
n
X
t=1
x1t
!( 1 )
2
is negative if (2) holds but positive if
>1
:
=1
;
When
the price received by each …rm s is una¤ected by the amount supplied by any other
…rm, since in that case, (1) reduces to
ps (x1 ; : : : ; xn ) = mxs
Bertrand, Cournot and Monopolistically Competitive Equilibria
7
Remark 2. It is useful to have an expression for the price received by each …rm
when all …rms supply the same amount. If we evaluate the inverse demand function
(1) at (x1 ; : : : ; xn ) = (x; : : : ; x) the result is
p (x)
ps (x; : : : ; x) = mn( 1 ) 1 x
1
:
This is the inverse of what Dixit and Stiglitz call ”the Chamberlinian DD curve.”
The demand elasticity of this inverse demand curve is 1 1 :
Remark 3. The monopoly case occurs when there is only one …rm. In this case, the
monopolist faces the inverse demand curve
p1 (x1 ) = mx1 1 ;
which is the special case of (1) obtained when n = 1: The demand elasticity of
this inverse demand curve is also 1 1 : Thus, the elasticity of the monopoly inverse
demand curve is the same as the elasticity of the inverse of the Chamberlinian DD
curve mentioned in Remark 2.
Remark 4. If we de…ne
H
n
X
x1t
t=1
then we can rewrite (1) as
!( 1 1 )[
(1
)]
(3)
;
(4)
ps (xs ) = mxs H:
By following Dixit and Stiglitz and assuming that, because n is large, the supply xs
chosen by …rm s has virtually no e¤ect on H; we can treat H as a constant. We
will call the inverse demand curve ps (xs ) described in (4) the Dixit-Stiglitz inverse
demand curve. It is, in fact, the inverse of what Dixit and Stiglitz refer to as ”the dd
curve” of ”Chamberlinian terminology.” The elasticity of the Dixit-Stiglitz inverse
demand curve (4) is 1 when H is independent of xs :
The Bertrand Case:. For the purpose of investigating the Bertrand equilibrium it is necessary to describe the demand function. Fortunately, it is straightforward to invert the inverse demand functions given in (1) to obtain the corresponding
demand function faced by …rm s: The result is
![ (1 ) ][ (1 1 ) 1 ]
n
X
1
1
1
xs (p1 ; : : : ; pn ) = m (1 ) ps
(5)
pt
t=1
= m
1
1
(1
)
ps
n
X
t=1
1
pt
"
1
! (1 (1 ) )
1
(1 1 )
1
#
:
8
Bertrand, Cournot and Monopolistically Competitive Equilibria
Remark 5. Note that
@xs (p1 ; : : : ; pn )
@ps
=
2 0
xs (p1 ; : : : ; pn ) 4 1 @
1
ps
is negative if both and 1
However, when t 6= s;
1
@xs (p1 ; : : : ; pn )
= m (1
@pt
1
ps
1
1
Pn
1
t=1 pt
A+
1
1
ps
1
) Pn p1
t=1 t
(1
3
1
5
1
are positive. This is true even if (2) fails.
)
1
1
(1
1
1
pt ps
)
n
X
"
1
! (1 (1 ) )
1
(1 1 )
1
pt
t=1
2
#
is positive if (2) holds but negative if
>1
:
=1
;
When
the amount sold by each …rm s is una¤ected by the price charged by any other …rm.
Remark 6. Again, it will be useful in what follows to have an expression for the
demand of each …rm when all …rms make the same choice. In this case, that means all
…rms charge the same price. If we evaluate the demand function (5) at (p1 ; : : : ; pn ) =
(p; : : : ; p) the result is
x (p)
xs (p; : : : ; p) = m (1
1
)
p
1
(1
)
n[ (1
)
][ (1 1
1
)
]:
The demand function x (p) is, of course, the inverse of p (x) de…ned in Remark 2;
it is the ”Chamberlinian DD curve.” As asserted in Remark 2, the elasticity of this
demand curve is 1 1 :
Remark 7. When there is only one …rm, that …rm is a monopolist who faces the
demand curve
1
1
x1 (p1 ) = m (1 ) p1 (1 )
which is easily seen to be the inverse of p1 (x1 ) ; the inverse demand curve described
in Remark 3: As noted in that Remark, the elasticity of this demand curve is 1 1 and
is the same as the elasticity of the Chamberlinian DD curve mentioned in Remarks
2 and 6.
Bertrand, Cournot and Monopolistically Competitive Equilibria
9
Remark 8. Let’s de…ne
K
n
X
1
pt
t=1
and rewrite (5) as
![ (1
xs (ps ) = m (1
)
][ (1 1
1
)
]
;
(6)
1
1
)
ps K:
(7)
We will again follow Dixit and Stiglitz and treat K as a constant. We, thereby,
e¤ectively assume that, because n is large, the price ps chosen by …rm s has virtually
no e¤ect on K: The elasticity of the demand curve (7), that we refer to as the DixitStiglitz demand curve, is 1 because K is independent of ps : The Dixit-Stiglitz demand
curve is, of course, the inverse of ps (xs ) ; of Remark 4. It is, therefore, the dd curve
of Chamberlinian terminology.
Interpreting the Parameters. As noted in the introduction, we will, at certain points in the analysis, consider cases in which market demand is ”large.” Note
that, for all s and for each price vector (p1 ; : : : ; pn ) ; an increase in the parameter m
increases the demand of …rm s in (5). Also as m becomes large the demand of …rm
s becomes large. For this reason we use m as the parameter that measures the
size of market demand. Thus, the case in which market demand is large is the
case in which m is large.
We also noted in the introduction that we would be required to impose a restriction
on the elasticity of demand. Speci…cally, we asserted that we would assume that the
elasticity of demand in the Dixit-Stiglitz monopolistically competitive equilibrium is
higher than in the monopoly case in which only one …rm produces. We discussed the
monopoly case in Remarks 3 and 7 and the Dixit-Stiglitz case in Remarks 4 and 8. In
those Remarks we observed that the elasticity of demand in the monopoly case
is 1 1 and that the elasticity of demand in the Dixit-Stiglitz monopolistically
competitive equilibrium is 1 : Clearly, condition (2) implies that
1
1
;
(8)
1
Thus, the case in which condition (2) holds is exactly the case in which
the elasticity of demand in the Dixit-Stiglitz monopolistically competitive
equilibrium is greater than the elasticity of demand in the monopoly case.
As noted in Remarks 2-4 and Remarks 6-8, the monopoly elasticity is the same as
the elasticity of the Chamberlinian DD curve and the Dixit-Stiglitz demand curve
is the Chamberlinian dd curve. As a consequence, condition (8) can also be
interpreted as imposing what Dixit and Stiglitz call the ”conventional”
Chamberlinian condition that the dd curve is more elastic than the DD
curve.
>
Bertrand, Cournot and Monopolistically Competitive Equilibria
10
Utility Maximization and Demand. It is straightforward to observe that,
when < 1; the demand function we propose to study is the utility maximizing demand function of a representative consumer who consumes one other good in addition
to the n di¤erentiated products. In making this interpretation, we let (x1 ; : : : ; xn )
denote the vector of amounts consumed of the n di¤erentiated products and we use
y to denote the amount consumed of the one other good. When the representative
consumer faces the price vector (p1 ; : : : ; pn ) ; has income I and maximizes the utility
function
2
!( 1 ) 3
n
X
m
5
u (y; x1 ; : : : ; xn ) = 4y +
(9)
x1t
t=1
subject to the budget constraint
y+
n
X
pt xt = I;
t=1
his demand function is given by (5).
The condition < 1 is imposed to insure that the utility function (9) is quasiconcave. It is easy to verify that, whenever < 1;
m
n
X
t=1
x1t
!( 1 )
is a strictly concave function of (x1 ; : : : ; xn ) : This is then easily seen to imply that
the utility function (9) is, indeed, a quasi-concave function of (y; x1 ; : : : ; xn ) : Before
leaving this point, it should be emphasized that condition (2) is not required to
guarantee the quasi-concavity of the utility function (9), all that is required is < 1:
Remark 9. The utility function (9) is, of course, is not quite a special case of the
general utility function
0
!( 1 1 ) 1
n
X
A
u (y; x1 ; : : : ; xn ) = v @y;
x1t
t=1
assumed by Dixit and Stiglitz. The di¤erence arises because they assumed that v ( ; )
was homothetic, a condition not satis…ed by the function
v (y; z) = y + m
z
Bertrand, Cournot and Monopolistically Competitive Equilibria
11
used to de…ne the utility function (9). Nevertheless, the interpretation of is the
same here as in the Dixit-Stiglitz paper; viz, In addition to being the elasticity of
the demand curve faced by each …rm in monopolistic competition, 1 measures the
elasticity of substitution between goods s and t.
Remark 10. Dixit and Stiglitz imposed a condition analogous to condition (2) by
e¤ectively assuming that the elasticity of substitution between y and z was smaller
than 1 ; the elasticity of substitution between the di¤erentiated goods. The interpretation of that assumption in their analysis is the same as the interpretation of (2)
in ours. In Chamberlinian terms, each of these assumptions guarantee that the dd
curve is more elastic than the DD curve.
Demand Elasticity. We can obtain the elasticity of demand by using either
the demand function (5) or the inverse demand function (1). Let’s begin by using
the demand function (5) and the expression for
@xs (p1 ; : : : ; pn )
@ps
derived in Remark 5. The resulting expression for the elasticity is
"bs
(p1 ; : : : ; pn )
1 ;:::;pn )
ps @xs (p@p
s
x (p ; : : : ; pn )
2 s0 1
1
1@
ps
4
=
1
Pn
(10)
1
1
t=1 pt
1
1
A+
1
ps
1
1
) Pn p1
t=1 t
(1
1
3
5:
Using the inverse demand function (1) and the expression for
@ps (x1 ; : : : ; xn )
@xs
given in Remark 1, we obtain another expression for the elasticity of demand. Specifically,
ps (x1 ; : : : ; xn )
"cs (x1 ; : : : ; xn )
(11)
1 ;:::;xn )
xs @ps (x@x
s
= h
1
1
xs1
Pn
t=1
x1t
+ (1
)
x1s
Pn
t=1
x1t
Interpreting The Expressions for the Demand Elasticities.
i:
Bertrand, Cournot and Monopolistically Competitive Equilibria
12
The Bertrand Case. The important features of the expression for the demand
elasticity "bs (p1 ; : : : ; pn ) given in (10) are described in the following series of remarks.
Remark 11. We have already noted in Remarks 3 and 7 that when there is only one
…rm, the elasticity of the demand curve faced by that …rm is 1 1 : The same result
follows from (10), since, when n = 1;
1
1
ps
Pn
and (10) reduces to
"b1 (p1 ) =
=1
1
1
t=1 pt
1
(1
)
:
Remark 12. When n > 1; expression (10) tells us that the elasticity of demand
faced by a typical …rm s, "bs (p1 ; : : : ; pn ) ; is a weighted average of 1 and (1 1 ) in
which
1 1
ps
0< P
<1
1 1
n
p
t
t=1
is the weight on the monopoly elasticity,
1
(1
1
)
; and
1
ps
0<1
Pn
1
t=1 pt
1
<1
is the weight on the monopolistically competitive elasticity, 1 :
The subsequent remarks are simple consequences of the one just made.
Remark 13. When n > 1 and condition (2) holds,
1<
1
(1
)
< "bs (p1 ; : : : ; pn ) <
1
;
(12)
i.e., the elasticity of a typical …rm s, "bs (p1 ; : : : ; pn ) ; is less than the monopolistically
competitive elasticity but above the monopoly elasticity.
Bertrand, Cournot and Monopolistically Competitive Equilibria
Remark 14. Note that for
n
X
!
1
1
pt
t=1
…xed,
2
1
1
6
=6
4
ps
Pn
1
t=1 pt
1
13
ps
Pn
1
t=1 pt
1
1
1
1
31
1
7
7
5
:
is a decreasing function of ps : Thus, as …rm s raises its price, ps ; relative to the
geometric average of all …rms’prices
n
X
1
1
pt
t=1
!
1
1
1
;
it’s elasticity of demand, "bs (p1 ; : : : ; pn ) ; rises and is closer to the monopolistically
competitive elasticity 1 :
Remark 15. Note also that even if we allow
n
X
1
pt
1
t=1
!
to vary as ps increases
1
1
ps
Pn
1
t=1 pt
1
=
1
1 + ps
is clearly a decreasing function of ps : Thus,
1
P
1
1
t6=s pt
1
;
@ebs pb1 ; : : : ; pbn
<0
@ps
and as …rm s raises its price, ps ; it’s elasticity of demand, "bs (p1 ; : : : ; pn ) ; rises and is
closer to the monopolistically competitive elasticity 1 :
Remark 16. In the special case that arises when all …rms charge the same price, p;
"bs (p1 ; : : : ; pn ) reduces to
"b (n)
"bs (p; : : : ; p) =
1
1
1
n
+
1
(1
1
:
)n
Bertrand, Cournot and Monopolistically Competitive Equilibria
14
Remark 17. When there are many …rms and
n
X
1
pt
t=1
1
!
1
1
1
is very large relative to ps ; then
1
1
ps
Pn
1
'0
1
t=1 pt
and (10) tells us that
"bs (p1 ; : : : ; pn ) '
1
:
This means that the demand curve facing each …rm has an elasticity near that faced
by …rms in Dixit and Stiglitz’s model of monopolist competition. In particular,
"b (n) =
1
1
1
n
1
+
(1
1
1
'
)n
when n is large and all …rms charge the same price.
Note that the comments made in this Remark are true whether (2) holds or not.
The Cournot Case. Once again, we list the important features of the expression for the demand elasticity "cs (x1 ; : : : ; xn ) given in (11) in a series of remarks.
Remark 18. We already know from Remarks 3, 7 and 11 that the expression for
"c1 (x1 ) given in (11) must reduce to
"c1 (x1 ) =
1
(1
)
:
when n = 1: This, also follows from (11) since
x1
Pn s
t=1
when n = 1.
x1t
= 1;
Remark 19. When n > 1; expression (11) tells us that
1
;
"cs (x1 ; : : : ; xn )
Bertrand, Cournot and Monopolistically Competitive Equilibria
15
the inverse of the elasticity of demand faced by a typical …rm s, is a weighted average
of the inverse elasticities and (1
) in which
x1
0 < Pn s
t=1
is the weight on, (1
<1
x1t
) ; the inverse of the monopoly elasticity,
x1
Pn s
0<1
t=1
x1t
1
(1
)
; and
<1
is the weight on, ; the inverse of the monopolistically competitive elasticity, 1 :
The following remarks are simple consequences of the one just made.
Remark 20. When n > 1; and condition (2) holds,
<
"cs
and
1<
1
<1
(x1 ; : : : ; xn )
1
1
<1
< "cs (x1 ; : : : ; xn ) <
1
(13)
;
i.e., the elasticity of a typical …rm s, "cs (x1 ; : : : ; xn ) is less than the monopolistically
competitive elasticity but above the monopoly elasticity.
Remark 21. Note that for
n
X
x1t
t=1
…xed,
x1
Pn s
t=1
x1t
2
=4 P
n
!
xs
1
1
t=1 xt
1
31
5
:
is an increasing function of xs : Thus, as …rm s raises its supply, xs ; relative to the
geometric average of all …rms’supplies
n
X
t=1
x1t
!1 1
;
Bertrand, Cournot and Monopolistically Competitive Equilibria
16
it’s elasticity of demand "cs (x1 ; : : : ; xn ) falls and is closer to the monopoly elasticity
1
: Alternatively we could observe that, if the competition faced by …rm s increases
1
in the sense that xs falls relative to
n
X
x1t
t=1
!1 1
;
then "cs (x1 ; : : : ; xn ) increases and is closer to monopolistically competitive lower bound
1
:
Remark 22. In the special case that arises when all …rms supply the same amount,
(11) reduces to
"c (n)
"cs (x; : : : ; x) =
1
+ (1
1
n
1
) n1
:
Remark 23. When there are many …rms and
n
X
x1t
t=1
!1 1
is very large relative to xs ; then
x1
Pn s
t=1
and (11) tells us that
x1t
'0
"cs (x1 ; : : : ; xn ) '
1
:
This means that the demand curve facing each …rm has an elasticity near that faced
by …rms in Dixit and Stiglitz’s model of monopolist competition. In Particular,
"c (n) =
1
1
n
1
+ (1
) n1
'0
when n is large all …rms supply the same amount.
Again, the comments made in this Remark are true whether (2) holds or not.
Bertrand, Cournot and Monopolistically Competitive Equilibria
17
Comparing the Bertrand, Cournot and Dixit-Stiglitz Elasticities.
Remark 24. When n = 1;
"c1 (x1 ) = "c (1) =
1
(1
)
= "b (1) = "b1 (p1 ) :
When n > 1 and all …rms supply the same amount and charge the same price,
"c (n) < "b (n) :
These results are true whether or not condition (2) holds.
Proof: The case n = 1 was discussed in Remarks 3, 7, 11 and 18.
Since
1
f (x) =
x
is convex, Jensen’s inequality implies that
"c (n) =
=
<
=
=
1
1
+ (1
) n1
1
1
f
1
+ (1
)
n
n
1
1
f ( ) + f (1
)
1
n
n
1
1
1
1
1
+
n
(1
)n
b
" (n) : k
1
n
Remark 25. Clearly, if (2) failed and we instead had
0<1
<
< 1;
(14)
then 1 ; the elasticity of the demand curve faced by …rms in monopolistic competition,
would actually be less than the elasticity of the demand curve faced by a monopolist
who faced no such competition. Condition (14) would also imply that when …rms
faced ”competition”from other …rms producing ”similar”but di¤erentiated products
the elasticity of the demand curve (5) they faced would be lower than that faced by
a monopolist. Furthermore, if (14) holds, then "bs (p1 ; : : : ; pn ) ; the demand elasticity
faced by …rm s; would actually decrease and move closer to 1 when the ”competition”
faced by …rm s increased in the sense that ps rose relative to
! 11
n
1
X
1
1
pt
:
t=1
Bertrand, Cournot and Monopolistically Competitive Equilibria
18
In addition, (14) implies that "cs (x1 ; : : : ; xn ) ; also decreases and moves closer to 1
when the ”competition”faced by …rm s increases in the sense that xs falls relative to
n
X
!1 1
x1t
t=1
The restriction (2) on the parameters
and
:
obviously rules out such cases.
Remark 26. Finally, note that if (2) fails and we instead have
1
= ;
then
"bs (p1 ; : : : ; pn ) = "cs (x1 ; : : : ; xn ) =
1
=
1
1
:
We will discuss this borderline case at some length in Section 10 at the end of the
paper.
3.2. Costs. All …rms will be assumed to face the same total cost function C (x) :
We will consider two di¤erent cases: the case of increasing marginal cost and Ushaped average cost curves and the Dixit-Stiglitz case in which the marginal cost is
constant and there is a …xed cost. For the most part the analysis of these two
cases is the same, but there are some points at which it is useful to distinguish the
cases. Also it is often possible to be more speci…c about the Dixit-Stiglitz case. In
particular, the Monopolistically Competitive, Cournot and Bertrand equilibria can
all be explicitly computed for this case.
Cost Case 1: The Case of U-Shaped Average Cost. In this case, the
marginal cost function
M C (x) = C 0 (x)
will be assumed to be increasing and the average cost function
AC (x) =
C (x)
x
will be assumed to U-shaped. We will also assume that C 00 (x) exists and is continuous
and that
lim AC (x) = 1:
x!0
In this case, as in Cost Case 2, there are both …xed and variable costs and the marginal
costs are the marginal variable costs.
Bertrand, Cournot and Monopolistically Competitive Equilibria
19
Cost Case 2: The Dixit-Stiglitz Case. Dixit-Stiglitz assumed that
C (x) = F + cx
where F > 0 and c > 0: In this case,
M C (x) = C 0 (x) = c
and
F
C (x)
= + c:
x
x
So marginal cost is constant and average cost is always declining and larger than
marginal cost.
AC (x) =
4. The Chamberlin-Dixit-Stiglitz Equilibrium
This section simply summarizes Dixit and Stiglitz’s results as they relate to the
present model. Let’s begin by recalling the discussion in Remark 8 and in the subsection on Utility Maximization and Demand. As a result of that discussion we know
that, when demand is derived from the utility function (9), the demand curve faced
by each …rm in Dixit and Stiglitz’s model of monopolistic competition is
xs (ps ) = m (1
1
1
)
ps K
where K; which is in fact related to ps by (6), is treated as independent of ps . Equivalently, we can, as noted in Remark 4, derive the Chamberlin-Dixit-Stiglitz equilibrium,
using the inverse demand curve
ps (xs ) = mxs H
and treat H; which is, in fact, related to xs by (3), as independent of xs . Except
for the special form of the constant mH multiplying xs ; this is exactly the demand
function derived by Dixit and Stiglitz.
De…nition 1. In the Dixit Stiglitz equilibrium each …rm produces
x ( ) = arg max [Rs (xs )
C (xs )]
xs
where (4) implies that the revenue function of …rm s is
Rs (xs ) = ps (xs ) xs = mx1s
H
and H is treated as being independent of xs : Since the price received by each …rm is
p( )
ps (x ( )) = AC (x ( )) ;
(15)
Bertrand, Cournot and Monopolistically Competitive Equilibria
20
no …rm makes a pro…t. When each of the n ( ) …rms produces x ( ) ; (3) becomes
( 1 1 )[
n ( ) x ( )1
1)
= n ( )( 1
x ( )[ (1
H =
(1
)]
)]
:
and
p ( ) = mx ( )
becomes
1
p ( ) = mx ( )
H
n ( )( 1
1)
;
(16)
which is the condition that determines the number of …rms, n ( ). The Dixit-Stiglitz
equilibrium is completely described by the output, x ( ) ; of each …rm, the price,
p ( ) ; received by each …rm and the number of …rms, n ( ) :
Remark 27. In both Cost Cases 1 and 2, x ( ) is determined by
C 0 (x ( )) = (1
So that
p( ) =
) AC (x ( )) :
C 0 (x ( ))
(1
)
(17)
(18)
The number of …rms is
"
#
mx ( ) 1 (1
n( ) =
AC (x ( ))
1
1
)
:
(19)
Proof: When, as is true in both cases one and two,
C 00 (xs )
0;
the maximand
[Rs (xs )
C (xs )]
is strictly concave and x ( ) is obtained as the solution to
(20)
M Rs (x ( )) = M C (x ( ))
where the marginal revenue is
M Rs (xs ) = (1
) mxs H = (1
) ps (xs ) :
(21)
Together (15), (20) and (21) imply (17). Equation (18) follows from (15) and (17).
Substituting (15) in (16) and solving for n ( ) we get (19).k
Bertrand, Cournot and Monopolistically Competitive Equilibria
Remark 28. Since
=
21
1
demand elasticity
the relationship between p ( ) and the marginal cost in (18) is a standard result which
asserts that
marginal cost
i:
price = h
1
1 demand elasticity
Remark 29. Note that in Cost Case 1,
x (0) = argminAC (x)
x
is the competitive output of each …rm and the competitive outcome is the Chamberlinian outcome. When > 0; (17) implies that
M C (x ( )) < AC (x ( ))
so that
x ( ) < x (0) ;
i.e., the monopolistically competitive output of each …rm is less than the average cost
minimizing competitive level x (0) :
The equilibrium is described graphically by Figure 1 which is familiar from textbook expositions of Chamberlin’s model. Figure 1 is obviously drawn for Cost Case
1.
4.1.
When Marginal Cost is Constant.
Remark 30. In this case, we can explicitly compute the Dixit-Stiglitz monopolistically competitive equilibrium. In particular, (17) and (18) imply that, in equilibrium,
each …rm supplies
1
F
x( ) =
1
c
and charges
p( ) =
c
(1
)
:
Bertrand, Cournot and Monopolistically Competitive Equilibria
22
MC
AC
p (α )
= AC (x(α ))
Demand
MC(x(α ))
= (1 − α )AC (x(α ))
MR
x(α )
x(0 )
Using (19), we observe that the number of …rms is
"
# 1
mx ( ) 1 (1 1 )
n( ) =
AC (x ( ))
2
h
i
(1
) F
m
c
6
= 4(1
)
c
=
"
m
(1
)
F
1
c
#
(22)
1
3
7
5
1
(1
1
)
1
(1
1
)
:
4.2. The Size of the Market, Price, Output Per Firm and the Number of
Firms.
Remark 31. Note that, since x ( ) is simply determined by condition (17) which
only involves the cost function and the demand parameter , x ( ) is independent of
other aspects of demand and, in particular, of the size of the market parameter m.
Since condition (15) asserts that the price is
p ( ) = AC (x ( )) ;
Bertrand, Cournot and Monopolistically Competitive Equilibria
23
it also is independent of the size of the market.
Remark 32. The expression for the equilibrium number of …rms given in (19) implies
that the number of …rms is a¤ected by the size of demand. It is also clear from
expressions (19) and (22) that the equilibrium number of …rms, n ( ) ; grows with
the market size parameter m when and only when condition (2) holds so that
1
>
1
1
and the elasticity of demand in monopolistic competition is higher than in the monopoly
case.
If this elasticity assumption fails and we have
1
<
1
;
1
then n ( ) is a decreasing function of m. In the case where
1
=
1
;
1
we cannot solve (19) to get (22). In that case,
H=K=1
and (16) becomes
p ( ) = mx ( )
1
= AC (x ( ))
which is independent of n ( ) : In this particular case, the number of …rms in monopolistic competition is indeterminate.
How the Firm’s Price A¤ects Demand when the Market is Large. Let’s
ask, in particular, what the size of the market means for the Dixit-Stiglitz assumption
that K and H are independent of the actions taken by any …rm. Let’s …rst suppose
that …rm s charges ps while all other …rms all charge p ( ) : Substituting in (6) we
get
1
K = (n ( )
1
1)[ (1 ) ][ (1
)
1]
ps
n( )
1
+ p( )
1
![ (1
)
][ (1 1
1
)
]
:
Alternatively, let’s suppose that …rm s supplies xs while all other …rms supply x ( ) :
Then (3) becomes
Bertrand, Cournot and Monopolistically Competitive Equilibria
H = (n ( )
1)
1
x1s
n( )
1
1
( 1 1 )[
+ x( )
1
(1
24
)]
:
If condition (2) holds then n ( ) is large when m is large and
K ' (n ( )
1
1)[ (1 ) ][ (1
)
1]
1
1
p ( )[
(1
)
]
(23)
while
H ' (n ( )
1
1) 1
x ( )[
(1
)]
:
(24)
This means that in a large market K is, indeed, approximately independent of ps and
H is approximately independent of xs as assumed by Dixit and Stiglitz. But this
argument applies only to the case in which the demand elasticity is higher
in monopolistic competition than in monopoly. If condition (2) fails and n ( )
is a decreasing function of m because
1
<
1
1
then the number of …rms will be small when the market size parameter is large. In
that case, there is no reason for (23) or (24) to hold in a large market.
The case in which
1
1
=
;
1
is special. We noted above that, in that case,
K=H=1
and the number of …rms in monopolistic competition is indeterminate. Thus, in that
case, the Dixit-Stiglitz assumptions that K is approximately independent of ps and
H is approximately independent of xs hold in the strongest possible sense. In that
case, K is completely independent of ps ; and H is completely independent of xs : This
is true whether the market is large or small.
4.3.
Comparative Statics of x ( ), p ( ) and n ( ).
Remark 33. As 1 ; the elasticity of demand, rises the output, x ( ) ; produced by
each …rm in the Dixit Stiglitz equilibrium rises and the price, p ( ) falls. These
results are illustrated in Figure 2. In Figure 2, < 0 , so that 10 < 1 and
AC (x ( ))
1
=
M C (x ( ))
1
<
1
1
0
=
AC (x ( 0 ))
:
M C (x ( 0 ))
Bertrand, Cournot and Monopolistically Competitive Equilibria
25
MC
AC
p(α ')
= AC (x(α '))
p(α )
= AC (x(α ))
MC (x(α ))
= (1 − α )AC(x(α ))
MC (x(α '))
= (1 − α ')AC (x(α '))
x(α ')x(α )
x(0)
Proof: Implicitly di¤erentiating (17) we get
x0 ( ) =
[C 00
AC (x ( ))
(x ( )) (1
) AC 0 (x ( ))]
Since
C 0 (x)
AC (x) =
x
0
C(x)
x
(25)
;
(17) implies that
AC 0 (x ( )) =
AC (x ( ))
< 0;
x( )
(26)
and the expression for x0 ( ) in (25) is negative.
Note that (15) implies that
p0 ( ) = AC 0 (x ( )) x0 ( ) :
(27)
Since we have shown that x0 ( ) and AC 0 (x ( )) are negative, the expression for p0 ( )
in (27) is positive.
In Cost Case 2, the result follows immediately from the expressions
x( ) =
1
1
F
c
Bertrand, Cournot and Monopolistically Competitive Equilibria
and
26
c
p( ) =
(1
)
obtained in Remark 30. These expressions imply that
1 F
<0
2 c
x0 ( ) =
and
p0 ( ) =
c
)2
(1
:k
Remark 34. Assume that m and, hence,
1
m
n( ) =
AC ( ) x ( )1
(1 )
1
are large. In that case, increases in 1 ; the elasticity of demand, lead to a reduction
in n ( ) the number of …rms that produce in the Dixit-Stiglitz equilibrium. This is a
surprising result, since we expect the e¤ect of an elasticity increase to be an increase
the number …rms.
Proof: We rewrite the expression for n ( ) in (19) as
m
( )
n( ) =
1
(1 )
1
where
( ) = AC ( ) x ( )1
:
Di¤erentiating, we get
n0 ( ) =
"
+
=
where
"
0
( )
1
1
0
1
1
( )
( )
m
( )
2 log
1
1
1
#
(1
1
1
m1 (1 ) ( )(1 )
)2
(1
1
"
#
m
( )
)
1
m
( )
1
1
(28)
1
1
1
1
2
log
(1 )
:
m
( )
#
:
(1 )
Bertrand, Cournot and Monopolistically Competitive Equilibria
27
When condition (2) holds, we have
1
1
>0
1
and
m
( )
n( ) =
implies that
1
1
(1 )
>1
m
>1
( )
so that
m
( )
log
> 0:
Thus,
)2
(1
1
1
the second term in the expression for
0
m
( )
log
;
; is negative. The term
( ) = p0 ( ) x ( )1
+ (1
)p( )x( )
is positive, however. This means that the sign of the term
ous. However, when m and, hence
n( ) =
are large enough
m
( )
and of n0 ( ) is ambigu-
1
1
(1 )
is negative so that the expression for n0 ( ) in (28) is positive.k
5. Cournot Equilibrium
At this point we …x the number of …rms at n:
De…nition 2. In a Cournot equilibrium, …rm s faces the inverse demand function
(1) and supplies the pro…t maximizing output level xcs which is de…ned formally as
xcs = argmax [Rs (xc1 ; : : : ; xs ; : : : ; xcn )
C (xs )]
xs
where (1) implies that
Rs (x1 ; : : : ; xn ) = ps (x1 ; : : : ; xn ) xs
=
mx1s
n
X
t=1
x1t
!( 1 )
1
:
Bertrand, Cournot and Monopolistically Competitive Equilibria
28
We identify the Cournot equilibrium with the vector, (xc1 ; : : : ; xcn ) specifying the
pro…t maximizing amount produced by each of the n …rms.
Remark 35. When condition (2) holds, the Cournot equilibrium (xc1 ; : : : ; xcn ) is characterized by the n equations
ps (xc1 ; : : : ; xcn ) = h
C 0 (xcs )
(1
(xc )1
Pn s c 1
t=1 (xt )
) 1
+
(xc )1
Pn s c 1
t=1 (xt )
i
(29)
which, recalling expression (11) for the elasticity "cs (xc1 ; : : : ; xcn ) ; (29) can also be
written as
C 0 (xcs )
c
c
:
(30)
ps (x1 ; : : : ; xn ) =
1
1 "c xc ;:::;xc
s( 1
n)
Proof: The …rst order condition satis…ed at xcs is
@Rs (xc1 ; : : : ; xcn )
= C 0 (xcs )
@xs
(31)
where
@Rs (x1 ; : : : ; xn )
@xs
= mxs
n
X
t=1
x1t
(32)
!( 1
= ps (x1 ; : : : ; xn ) (1
2)
"
) 1
(1
)
n
X
x1t
t=1
x1s
Pn
1
t=1 xt
+
x1s
!
x1
Pn s
#
+ x1s
1
t=1 xt
:
Substituting (32) in the …rst order conditions (31) yields (29) which, using the de…nition of "cs (xc1 ; : : : ; xcn ) in (11) can also be written as (30).
29
Bertrand, Cournot and Monopolistically Competitive Equilibria
It is straightforward to verify that the maximand [Rs (xc1 ; : : : ; xs ; : : : ; xcn ) C (xs )]
is a strictly concave function of xs when condition (2) holds by demonstrating that
@ 2 Rs (x1 ; : : : ; xn )
@x2s
=
mxs
( +1)
n
X
(33)
!( 1
x1t
t=1
+(
n
X
)) mxs 2
2 (1
2)
"
x1t
t=1
+ (1
) mxs
2
n
X
!( 1
x1t
t=1
(1
)
n
X
x1t
!
x1s
t=1
!( 1
3)
"
(1
n
X
)
+
x1t
x1s
x1s
t=1
2)
#
!
+ x1s
is negative. To do this, we simply rewrite (33) by combining the last two terms to
get
@ 2 Rs (x1 ; : : : ; xn )
@x2s
=
mxs
( +1)
n
X
!( 1
xt1
t=1
) xs 2
m (1
0
@ 2
h
1
n
X
x1t
t=1
(1
)
2)
!( 1
"
(1
)
x1t
x1s
t=1
2)
Pn
n
X
1
t=1 xt
Pn
x1s
+ x1s
1
t=1 xt
i
!
+
x1s
#
1
1A :
The …rst term in this expression is clearly negative. The second term is also negative
because (1 ) < 1 implies that both
2
and
exceed one.k
h
(1
)
Pn
1
1
t=1 xt
Pn
t=1
x1s
x1t
+
x1s
i
#
Bertrand, Cournot and Monopolistically Competitive Equilibria
30
In the next section we will describe the symmetric Cournot equilibrium in which
all …rms supply the same amount. The following section demonstrates that the symmetric equilibrium is the only Cournot equilibrium.
5.1.
Symmetric Cournot Equilibrium.
Proposition 1. When condition (2) holds, there exists a symmetric Cournot equilibrium in which each …rm s produces
xcs = xc
and charges
pc = p (xc ) = mn( 1 )
1
1
(xc )
(34)
where xc is the solution to
mn( 1 )
1
(xc )
1
C 0 (xc )
(1
) 1 n1 +
C 0 (xc )
=
1 "c 1(n)
=
1
n
(35)
:
The Cournot price pc can also be obtained as the solution to
C 0 (x (pc ))
(1
) 1 n1 +
C 0 (x (pc ))
=
1 "c 1(n)
pc =
1
n
(36)
:
where because of Remark 6
xc = x (pc )
m (1
1
)
(pc )
1
(1
)
n[ (1
)
][ (1 1
1
)
]:
(37)
Proof: As noted in Remarks 2 and 22, when all n …rms supply the same amount
x ; they all charge (34) and the expression for the elasticity "cs (xc1 ; : : : ; xcn ) becomes
c
"cs (xc ; : : : ; xc ) = "c (n) =
1
1
n
1
+ (1
)
1
n
:
(38)
Substituting (34) and (38) in (30) we observe that xc is the solution to (35). A
solution clearly exists since the price
p (xc ) = mn( 1 )
1
(xc )
1
;
Bertrand, Cournot and Monopolistically Competitive Equilibria
31
on the left side of equation (35) grows without bound as xc approaches zero and
approaches zero as xc becomes large. The solution is unique since the p (xc ) is a
decreasing function of xc ; while marginal cost C 0 (xc ) is either increasing in xc (as in
Cost Case 1) or independent of xc (as in Cost Case 2):
Using the observation in Remark 6 we can invert p (x) to get (37) and rewrite (35)
as in (36).k
At this point we will not consider the question of whether the …rms make positive
pro…ts in the Symmetric Cournot equilibrium. We will presently discuss this issue at
length, however.
Remark 36. Note that the relationship between the Cournot price, pc ; and the marginal cost in (36) is
marginal cost
i:
price = h
1 demand 1elasticity
As we noted in Remark 28 this standard result also holds in the Dixit-Stiglitz case.
Remark 37. It is clear that xc and pc depend on m; n and : In the subsequent
discussion, it will often be useful to emphasize the dependence of xc and pc on m and
n by using the notation xc (m; n) and pc (m; n) to denote the Cournot equilibrium
output and price when there are n …rms and the market size parameter is m. When
there is no danger of confusion we will simply denote xc (m; n) and pc (m; n) by xc
and pc :
The Case when Marginal Cost is Constant.
Remark 38. When marginal cost is constant and equal to c;
x
and
c
"
cn1 ( 1 )
=
m (1
) 1 n1 +
3 11
2
1 (1 )
cn
i5
:
= 4 h
1
m 1 "c (n)
pc =
>
c
(1
) 1
c
(1
)
1
n
= p( ):
+
1
n
#
1
n
=
1
1
(39)
c
1
1
"c (n)
(40)
32
Bertrand, Cournot and Monopolistically Competitive Equilibria
In this case, the pro…t per unit can be easily calculated as
"
1
pc c = c
1
(1
) 1 n1 +
n
"
#
1
= c
= c
1
"c (n)
1
"c (n)
"c
1
(n)
#
1
1
Proof: In this case, (35) becomes
mn( 1 )
1
(xc )
1
=
c
(1
1
n
) 1
+
1
n
an equation in which the right side is independent of x:c The solution is given in (39)
Also (36) becomes (40).k
5.2. Uniqueness of the Cournot Equilibrium. In the previous section we
characterized the symmetric Cournot Equilibrium in which all …rms produce the
same amount. In this section we prove that this is the only Cournot equilibrium by
establishing the following Proposition.
Proposition 2. Assume that condition (2) holds. In a Cournot equilibrium all …rms
must produce the same amount.
Proof: Using the expression (1) for the inverse demand function we can rewrite
condition (29) as
m (xcs )
n
X
(xct )1
t=1
!( 1 )
1
=h
C 0 (xcs )
(1
) 1
which implies that for all s; xcs is the solution to
m (1
)
n
X
(xct )1
t=1
= C
0
(xcs ) (xcs )
+
(xcs )1
!( 1 )
[(1
)
(xc )1
Pn s c 1
t=1 (xt )
+
(xc )1
Pn s c 1
t=1 (xt )
i
1
(41)
]
n
X
t=1
(xct )1
!( 1 )
2
Bertrand, Cournot and Monopolistically Competitive Equilibria
If we can prove that, for
n
X
33
(xct )1
t=1
xcs
…xed, there is a unique
that solves (41), then all …rms must supply that amount
in the Cournot equilibrium. When
n
X
(xct )1
t=1
is …xed, the left hand side of equation (41) is independent of s as is
[(1
)
]
n
X
(xct )1
t=1
the positive coe¢ cient of (xcs )1
!( 1 )
2
;
on the right hand side of equation (41). Since, when
n
X
(xct )1
t=1
is …xed,
@ C 0 (xcs ) (xcs ) + (xcs )1
@xcs
= C 00 (xcs ) (xcs ) + C 0 (xcs ) (xcs )
1
+ (1
) (xcs )
> 0;
the right hand side of equation (41) is an increasing function of xcs . As a result the
xcs that solves equation (41) is, indeed, unique and the same for all …rms s: Thus, all
…rms must supply the same amount. k
5.3. Price and Supply Related to Market Size and the Number of Firms.
The proposition established in this section shows that either an increase in the size of
the market or a decrease in the number of …rms raise both the Cournot equilibrium
price and the per …rm output. It should be emphasized that these results depend
crucially on the assumption that the demand elasticity in monopolistic competition
exceeds that of monopoly.
Proposition 3. Assume that condition (2) holds. In that case, xc (m; n) and pc (m; n)
are increasing functions of the market size parameter m. Also, pc (m; n) is a decreasing function of the number of …rms, n. If, in addition, there are more than two …rms,
then xc (m; n) is a decreasing function of the number of …rms, n.
Bertrand, Cournot and Monopolistically Competitive Equilibria
34
Proof: We will proceed by treating n as well as m as continuous variables and
implicitly di¤erentiating the equations (35) and (36) that determine xc (m; n) and
pc (m; n) respectively. We will then be able to determine the signs of the resulting
c
c
c
c
derivatives @x
; @p ; @x
and @p
: The general arguments we give will apply to both
@m @m
@n
@n
Cost Cases 1 and 2. Since, in Cost Case 2, where marginal cost is constant, we have
derived exact expressions for xc (m; n) and pc (m; n) we could also verify that the
c
c
c
c
proposition holds in that Case by directly computing @x
; @p ; @x and @p
.
@m @m @n
@n
Let’s de…ne
(m; n)
(m; n) (n)
where
mn( 1
(m; n)
and
(n)
1
"c
1
= (1
(n)
1)
1
n
) 1
1
n
+
:
If we also de…ne the function F (x; m; n) by
F (x; m; n)
(m; n) x
1
C 0 (x) ;
then we can rewrite the condition (35) satis…ed at xc (m; n) as
F (xc ; m; n)
(m; n) (xc )
1
C 0 (xc ) = 0:
(42)
Implicitly di¤erentiating (42) we get
@xc
=
@n
=
Fn (xc ; m; n)
Fx (xc ; m; n)
c
n (m; n) (x )
(m; n) (
1) (xc ) 2
(43)
1
C 00 (xc )
and
@xc
=
@m
=
Fm (xc ; m; n)
Fx (xc ; m; n)
1
c
m (m; n) (n) (x )
:
(m; n) (
1) (xc ) 2 C 00 (xc )
Since
Fx (xc ; m; n) = (m; n) (
1) (xc )
and
Fm (xc ; m; n) =
m
(m; n) (xc )
1
= n( 1
2
C 00 (xc ) < 0;
1)
(xc )
1
> 0;
(44)
Bertrand, Cournot and Monopolistically Competitive Equilibria
35
@xc
> 0:
@m
Combining (34) and (35) we get
pc (m; n) =
C 0 (xc (m; n))
) 1 n1 +
(1
Thus,
1
n
(45)
:
c
C 00 (xc (m; n)) @x
@pc
@m
=
1
@m
(1
) 1 n +
> 0:
1
n
c
Expression (43) for @x
; expression (42) for F (xc ; m; n) and condition (44) combine
@n
to imply that Fn (xc ; m; n) and, hence,
@xc
@n
have the same sign as n (m; n) : For the purpose of computing
that (m; n) can be written as
) n( 1
(m; n) = m (1
1)
+ n( 1
n
(m; n) let’s observe
2)
(1
1
)
:
Thus,
n
(m; n) = m (1
=
m (1
)
1 n( 1
1
) 1
Condition (2) implies that
n
2)
n( 1
1
2)
+
2 n( 1
1
1+
2
1
(m; n) has the same sign as
' (n)
1+
1
n
2
1
Clearly,
' (1) = 1
but
' (2) =
1
2
>0
1
<0
1
and
'0 (n)
1
2
1
n2
< 0:
:
3)
1
1
:
n
1
Bertrand, Cournot and Monopolistically Competitive Equilibria
36
So
' (n) < ' (2) < 0
and
@xc
<0
@n
if there are more than 2 …rms.
Di¤erentiating (45) we get
@pc
=
@n
c
C 00 (xc (m; n)) @x
@n
1
(1
) 1 n1 +
n
C 0 (xc ) [
(1
)]
+
1
1
2
n (1
) 1 n +
n
2:
The …rst term is negative when n 2; and condition (2) implies the second term is
negative. Thus, we know that pc decreases with n; when n 2: In fact, we can also
show that this holds for n 1; by the following argument.
Clearly the demand function x (p) derived in Remark 6 depends on the parameters
m and n as well as on price. Let’s make that dependence explicit by writing
x (p; m; n) = m (1
1
)
p
1
(1
)
n[ (1
)
][ (1 1
1
)
]:
(46)
Observe that
@x (p; m; n)
=
@p
1
1
m( 1 ) n(
1
1
1
)[1 ( 1 )] p
1
1
1
(47)
<0
and
@x (p; m; n)
=
@n
(1
1
)
(1
1
)
1
m( 1 ) n[ (1
)
][ (1 1
1
)
] 1p
1
1
<0
(48)
if condition (2) holds. If we use the notation introduced in (46) to rewrite (36), the
condition that determines pc (m; n) ; it becomes
pc (m; n) =
C 0 (x (pc (m; n) ; m; n))
(1
)]
(1
) + n1 [
which we choose to rewrite as
(pc (m; n) ; m; n) = 0
(49)
Bertrand, Cournot and Monopolistically Competitive Equilibria
37
where
(p; m; n)
C 0 (x (p; m; n))
p
(1
)+
1
n
[
(1
)] :
(50)
Implicitly di¤erentiating (49) we get
(pc ; m; n)
c
p (p ; m; n)
@pc
=
@n
n
(51)
where (50) implies that
p
C 00 (x (p; m; n)) @x(p;m;n)
@p
(p; m; n) =
p
C 0 (x (p; m; n))
p2
and
n (p; n) =
C 00 (x (p; m; n)) @x(p;m;n)
@n
p
1
+ 2[
(1
)] :
n
These expressions together with (47), (48), (51) and condition (2) imply that
and
p
(p; n) < 0
n
(p; n) < 0
@pc
< 0: k
@n
5.4. Pro…tability, Market Size and the Number of Firms. In this Section
we establish three results. The …rst, which we state as a Corollary to Proposition
3, demonstrates that, when the Cournot per …rm output level, xc (m; n) ; is less than
the average cost minimizing output level, x (0) ; both per unit …rm pro…ts and …rm
pro…ts increase with increases in the market size parameter, m; and decrease when n;
the number of …rms, increases. The next result, Proposition 5, demonstrates, that,
for any n; all …rms earn a pro…t in the n-…rm Cournot equilibrium if the market size
parameter, m; is large enough. The …nal result obtained in this section, Proposition 6,
demonstrates that, for any m; all …rms lose money in the n-…rm Cournot equilibrium
if n is su¢ ciently large.
Bertrand, Cournot and Monopolistically Competitive Equilibria
38
Corollary 4. Assume that condition (2) holds. The per unit pro…t,
pc (m; n)
AC (xc (m; n)) ;
and pro…ts per …rm,
pc (m; n) xc (m; n)
C (xc (m; n)) ;
are both increasing in m if
xc (m; n) < x (0) :
The per unit pro…t,
pc (m; n)
AC (xc (m; n)) ;
and pro…ts per …rm,
pc (m; n) xc (m; n)
C (xc (m; n)) ;
are both decreasing in n; if there are more than two …rms and
xc (m; n) < x (0) :
In Cost Case 2 where average cost is always falling, the hypothesis
xc (m; n) < x (0)
is unnecessary.
Proof: When
xc (m; n) < x (0) ;
the increase in xc (m; n) caused by an increase in m causes AC (xc (m; n)) to fall and
the Proposition 3 asserts that pc (m; n) is increasing in m: Since per unit pro…ts,
pc (m; n)
AC (xc (m; n)) ;
and xc (m; n) both increase with m; pro…ts per …rm,
pc (m; n) xc (m; n)
C (xc (m; n)) = [pc (m; n)
AC (xc (m; n))] xc (m; n) ;
must also increase with m:
Similarly, when there are more than two …rms and
xc (m; n) < x (0) ;
the decrease in xc (m; n) caused by an increase in n causes AC (xc (m; n)) to rise and
the proposition asserts that pc (m; n) is decreasing in n: Since per unit pro…ts,
pc (m; n)
AC (xc (m; n)) ;
Bertrand, Cournot and Monopolistically Competitive Equilibria
39
and xc (m; n) both decrease with n; pro…ts per …rm,
pc (m; n) xc (m; n)
C (xc (m; n)) = [pc (m; n)
AC (xc (m; n))] xc (m; n) ;
must also decrease with n: k
Let’s …x the number of …rms at n: We now show that we can always make m large
enough so that we can have n …rms producing pro…tably at the Cournot outcome for
the given n: In particular, we will show that the Cournot equilibrium is pro…table for
all …rms if the market size parameter, m; exceeds or equals
m (n; ) = n(1
1
)
C 0 (x ( ))
i:
h
x ( ) 1 1 "c 1(n)
Proposition 5. Assume that condition (2) holds. If m = m (n; ) and there are n
…rms, then the Cournot equilibrium output is
xc (m (n; ) ; n) = x ( ) ;
the Cournot price charged by all …rms is
pc (m (n; ) ; n) = m (n; ) n( 1
1)
x( )
1
C 0 (x ( ))
i:
=h
1 "c 1(n)
(52)
and this price exceeds average cost so that all …rms earn a pro…t. This equilibrium
is described in Figure 3.
If m > m (n; ) and there are n …rms, then the Cournot equilibrium output is
xc (m; n)
x( );
the Cournot price charged by all …rms is
pc (m; n)
C 0 (x ( ))
i
h
1 "c 1(n)
and this price exceeds average cost so that all …rms earn a pro…t.
If m = m (n; ) and there are 2
n0 < n …rms, then the Cournot equilibrium
output is
xc (m; n0 ) x ( ) ;
the Cournot price charged by all …rms is
pc (m (n; ) ; n0 )
C 0 (x ( ))
h
i
1 "c 1(n)
Bertrand, Cournot and Monopolistically Competitive Equilibria
40
MC
AC
C ' (x(α ))
 1
1
γ 1 −  + (1 − α ) 
 n
n
C ' (x(α ))
=
1
1− c
ε (n )
= p c (m* (n, α ), n )
AC (x(α ))
C ' (x(α ))
=
1−α
Demand
C' (x(α ))
MR
(
x(α ) = x c m* (n, α ), n
)
x(0 )
and this price exceeds average cost so that all …rms earn a pro…t.
If m = m (n; ) and there are n0 > n
2 …rms, then the Cournot equilibrium
output is
xc (m (n; ) ; n0 ) x ( ) ;
the Cournot price charged by all …rms is
pc (m (n; ) ; n0 )
C 0 (x ( ))
i:
h
1 "c 1(n)
Proof: It is immediate to verify that m (n; ) has been chosen so that x ( )
satis…es condition (35) so that
xc (m (n; ) ; n) = x ( ) :
The Cournot equilibrium price in (52) is obtained by substituting x ( ) for xc in (35)
and combining (34) and (35). Since
C 0 (x ( )) = (1
) AC (x ( ))
Bertrand, Cournot and Monopolistically Competitive Equilibria
41
(52) implies
pc (m (n; ) ; n) =
(1
h
1
) AC (x ( ))
i
1
"c (n)
(53)
> AC (x ( )) :
The last inequality follows from the observation made in Remark 20 which implies
that
(1
)
h
i > 1:
1 "c 1(n)
If m > m (n; ) and there are n …rms, Proposition 3 implies that the Cournot
equilibrium output, xc (m; n) ; exceeds x ( ) and the Cournot price, pc (m; n) ; charged
by all …rms exceeds that given in (52). The fact that all …rms make a pro…t is a
consequence of Corollary 4 if
xc (m; n) < x (0) :
It is possible, however, that m is so large that xc (m; n) exceeds x (0) : In that case,
pc (m; n) > C 0 (xc (m; n))
AC (xc (m; n))
which means that all …rms make a pro…t.
If m = m (n; ) and there are 2 n0 < n …rms, Proposition 3 again implies that
the Cournot equilibrium output, xc (m (n; ) ; n0 ) ; exceeds x ( ) and the Cournot
price, pc (m (n; ) ; n0 ) ; charged by all …rms exceeds that given in (52). The fact
that all …rms make a pro…t is a also a consequence of the Corollary 4 if
xc (m (n; ) ; n0 ) < x (0) :
Again the possibility of
xc (m (n; ) ; n0 )
x (0)
exists and again …rms operate pro…tably in that case because then
pc (m (n; ) ; n0 ) > C 0 (m (n; ) ; n0 )
AC (m (n; ) ; n0 ) :
Finally, if m = m (n; ) and there are n0 > n …rms, the results that
xc (m (n; ) ; n0 )
x( )
and that the Cournot price, pc (m (n; ) ; n0 ) ; charged by all …rms is less than that
given in (52) are corollaries of Proposition 3.k
Now let’s …x the market size parameter m: The next proposition demonstrates
that, when is n large enough, all …rms lose money in the Cournot equilibrium.
Bertrand, Cournot and Monopolistically Competitive Equilibria
42
Proposition 6. For every m; there exists an n such that, when the market size is m
and there are n n; …rms
pc (m; n) < AC (xc (m; n))
so that all …rms lose money in the Cournot equilibrium.
Proof: For m and x; …xed let
0
n (m; x) = @
Note that
h
1
"c (n)
m 1
x1 C 0 (x)
i1
1
1
(1 )
A
xc (m; n (m; x)) = x
and
pc (m; n (m; x)) = m [n (m; x)]( 1 )
1
x
1
=h
C 0 (x)
1
1
"c (n)
i:
These equations hold even if n (m; x) is not an integer which it need not always be.
But if n (m; x) is an integer, and if there are n (m; x) …rms, then each …rm produces
xc (m; n (m; x)) = x
and charges pc (m; n (m; x)) in the Cournot equilibrium.
Also note that
lim n (m; x) = 1
x!0
and that n (m; x) is an increasing function of m and a decreasing function of x in
both Cost Cases 1 and 2. Finally, note also that
m (n (m; x ( )) ; ) = m:
Let’s …x
x < x( )
small enough so that
n (m; x)
2:
Then Proposition 3 applies to guarantee that
xc (m; n)
x < x( )
Bertrand, Cournot and Monopolistically Competitive Equilibria
43
when
n
n (m; x) :
Recall that, in Cost Case 1, marginal cost, C 0 (x) ; is always increasing while average
cost, AC (x) ; is decreasing when output is below
x ( ) < x (0)
and that, in Cost Case 2, marginal cost, C 0 (x) ; is constant and that average cost,
AC (x) ; is always decreasing. Thus, not only is
xc (m; n)
x < x( )
when
n
n (m; x)
but we also have
pc (m; n) =
C 0 (xc (m; n))
h
i
1
1 "c (n)
(54)
C 0 (x ( ))
h
i
1 "c 1(n)
and
AC (xc (m; n)) > AC (x) > AC (x ( )) :
Now let
=
AC (x)
(55)
AC (x ( ))
2
and choose
n > n (m; x)
su¢ ciently large so that
n
n
implies
<
C 0 (x ( ))
h
i
1 "c 1(n)
C 0 (x ( ))
+ :
(1
)
When
n
n;
(56)
Bertrand, Cournot and Monopolistically Competitive Equilibria
inequality (56) combines with the de…nition of
x ( ) to imply
44
and condition (17) that determines
C 0 (x ( ))
h
i
1
1 "c (n)
(57)
C 0 (x ( ))
+
(1
)
= AC (x ( )) +
<
= AC (x ( )) +
=
AC (x)
AC (x ( ))
2
AC (x) + AC (x ( ))
:
2
Since n has been chosen to exceed n (m; x) ; (54), (55) and (57) hold when
n
n:
Since (55) implies
AC (x) + AC (x ( ))
< AC (x) ;
2
(54), (55) and (57) combine to imply that
pc (m; n) < AC (xc (m; n))
when
n
n: k
When n is as de…ned in the proof of Proposition 6, the Cournot equilibrium price,
pc (m; n) and output, xc (m; n) for the case,
n
n
are illustrated in Figure 4.
6. Cournot-Free Entry Equilibrium
Now we assume that there is an in…nity of potential …rms all of which possess the
same cost function satisfying the conditions of either Cost Case 1 or Cost Case 2.
De…nition 3. In a Cournot free-entry equilibrium there are nc …rms that produce. The vector of amounts produced by these …rms, (xc1 ; : : : ; xcnc ) ; is a Cournot
Bertrand, Cournot and Monopolistically Competitive Equilibria
45
MC
AC
(
)
AC x c (m, n )
_
AC  x 
 
_
AC  x  + AC (x(α ))
 
2
p c (m, n )
AC (x(α ))
MC(x(α ))
= (1 − α )AC (x(α ))
Demand
MR
_
x
x c (m, n )
x(α )
x(0 )
equilibrium and each of these producing …rms makes a nonnegative (but possibly
zero) pro…t. Thus, for each s;
max [Rs (xc1 ; : : : ; xs ; : : : ; xcnc )
C (xs )]
xs
= [Rs (xc1 ; : : : ; xcnc )
C (xcs )]
0
where (1) implies that
Rs (x1 ; : : : ; xn ) = ps (x1 ; : : : ; xn ) xs
=
mx1s
n
X
x1t
t=1
!( 1 )
1
:
In addition, none of the non-producing …rms can make a pro…t. Thus, the …nal
condition satis…ed in a Cournot-Free Entry equilibrium is
max [Rs (xc1 ; : : : ; xcnc ; x)
x
C (x)]
0:
In analyzing the Cournot free-entry equilibria, we make use of fact established
in Proposition 2 that, for each n; the unique Cournot equilibrium is the symmetric
Bertrand, Cournot and Monopolistically Competitive Equilibria
46
Cournot Equilibrium shown to exist and described in Proposition 1. Of course, in
that symmetric equilibrium, all …rms produce the same amount xc (m; n) :
The proof that a Cournot free-entry equilibrium exists makes use of the following
Lemma which applies to a …rm that faces n rivals each producing x^: Note that (1)
implies that the price received by such a …rm will be
pn+1 (^
x; : : : ; x^; x) = mx
x1
+ n^
x1
( 1 1 )[
(1
)]
if it supplies x units.
Lemma 7. The pro…ts of a …rm who faces n rivals, each producing x^; is a decreasing
function of the amount, x^: Formally,
max [pn+1 (^
x; : : : ; x^; x) x
x
C (x)]
is a decreasing function of x:
Proof: The lemma follow immediately from the fact that pn+1 (^
x; : : : ; x^; x) is a
decreasing function of x^: k
Proposition 8. Assume that n
~ > 1: If m m (~
n; ) ; there exists a Cournot freeentry equilibrium with nc > n
~ …rms in which all …rms produce
xc (m; nc ) < x ( )
and charge a price
pc (m; nc ) > AC (x ( )) :
.
Proof: From Proposition 5 we know that if m m (~
n; ) ; then all …rms make a
pro…t in the n …rm Cournot equilibrium if 2 < n n
~ . In fact when m > m (~
n; ) ;
Proposition 5 implies that all …rms make a pro…t in the n …rm Cournot equilibrium
if 2 < n
n (m; x ( )) ; where n (m; x) is the function de…ned in the Proof of
Proposition 6. We have already noted that
m (n (m; x ( )) ; ) = m
which also implies that
n
~ = n (m (~
n; ) ; x ( ))
Since n (m; x) is increasing in m; m > m (~
n; ) implies
n (m; x ( )) > n
~:
Bertrand, Cournot and Monopolistically Competitive Equilibria
47
Proposition 6 tells us that for every m; there is some n; such that pro…ts are negative
if there are more than n …rms. There must therefore exist some nc 2 (n (m; x ( )) ; n)
for which
pc (m; nc ) AC (xc (m; nc ))
and
pc (m; nc + 1) < AC (xc (m; nc + 1)) :
We can demonstrate that the nc …rm Cournot equilibrium is, in fact, a Cournot
free-entry equilibrium by demonstrating that
max [Rnc +1 (xc (m; nc ) ; : : : ; xc (m; nc ) ; x)
x
C (x)] < 0:
But this follows immediately from Lemma 7 and Proposition 3 which tells us that
because nc > n (m; x ( )) n
~ 2;
xc (m; nc ) > xc (m; nc + 1) :
Finally note that, since nc > n (m; x ( )) ;
xc (m; nc ) < x ( )
and
pc (m; nc )
AC (xc (m; nc )) > AC (x ( )) : k
Proposition 9. When the market size parameter m is su¢ ciently large, the amount
supplied by each of the nc …rms in the Cournot free-entry equilibrium is
xc (m; nc ) ' x ( )
and the price charged by each of these …rms is
pc (m; nc ) ' AC (x ( )) :
Proof: In equilibrium,
(1
C 0 (xc (m; nc ))
) 1 n1c +
1
nc
= pc (m; nc )
Thus,
1
(1
) 1
1
nc
+
1
nc
AC (xc (m; nc )) :
AC (xc (m; nc ))
:
C 0 (xc (m; nc ))
(58)
Bertrand, Cournot and Monopolistically Competitive Equilibria
48
Clearly
lim n (m; x ( )) = 1:
m!1
So if we choose m large enough, then n (m; x ( )) and nc > n (m; x ( )) will also
be large. By choosing m large enough we can, therefore, be sure that
1
AC (x ( ))
+ =
+ >
0
c
C (x ( ))
(1
)
(1
1
) 1
1
nc
1
nc
+
Combining this inequality with (58) yields
AC (xc (m; nc ))
C 0 (xc (m; nc ))
>
AC (x ( ))
:
C 0 (xc ( ))
Thus, for m su¢ ciently large
xc (m; nc ) ' x ( )
and
pc (m; nc ) =
(1
C 0 (xc (m; nc ))
) 1 n1c +
' AC (x ( )) : k
1
nc
7. Bertrand Equilibrium
We …x the number of …rms at n:
De…nition 4. In a Bertrand equilibrium, …rm s faces the demand function (5)
and charges the pro…t maximizing price pbs which is de…ned formally as
pbs = argmax Rs pb1 ; : : : ; ps ; : : : ; pbn
ps
C xs pb1 ; : : : ; ps ; : : : ; pbn
:
where (5) implies that
(59)
Rs (p1 ; : : : ; pn ) = xs (p1 ; : : : ; pn ) ps
= m
1
(1
)
1
ps
1
n
X
t=1
1
pt
![ (1
)
][ (1
1
1
)
]
:
We identify the Bertrand equilibrium with the vector, pb1 ; : : : ; pbn specifying the
price charged by each of the n …rms.
Bertrand, Cournot and Monopolistically Competitive Equilibria
49
Remark 39. When condition (2) holds, the Bertrand equilibrium pb1 ; : : : ; pbn is
characterized by the n equations
ps = 0
@1
"
1
1
C 0 xs pb1 ; : : : ; pbn
!
1
ps
Pn
1
1
+
1
t=1 pt
1
ps
1
(1
)
Pn
t=1
1
1
pt
1
# 11 :
A
(60)
which, recalling the expression (10) for the elasticity "bs pb1 ; : : : ; pbn ; can also be
written as
C 0 xs pb1 ; : : : ; pbn
i
ps = h
(61)
1
1 "b (p1 ;:::;pn )
s
These equations are the Bertrand analogs of (29) and (30).
Proof: The …rst order condition satis…ed at pbs is
@
Rs pb1 ; : : : ; pbn
C xs pb1 ; : : : ; pbn
@ps
@Rs pb1 ; : : : ; pbn
@xs pb1 ; : : : ; pbn
=
C 0 xs pb1 ; : : : ; pbn
=0
@ps
@ps
where
(62)
@xs (p1 ; : : : ; pn )
@Rs (p1 ; : : : ; pn )
= ps
+ xs (p1 ; : : : ; pn )
@ps
@ps
Substituting this expression in (62) and using the fact, observed in Remark 5, that
@xs pb1 ; : : : ; pbn
<0
@ps
(63)
we get
@
Rs pb1 ; : : : ; pbn
C xs pb1 ; : : : ; pbn
@p
3
1
0s 2
b
b
@x pb ; : : : ; pbn
xs p1 ; : : : ; p n
5 C 0 xs pb1 ; : : : ; pbn A s 1
= @pbs 41 +
@xs (pb1 ;:::;pbn )
@ps
pbs
@ps
"
#
!
@xs pb1 ; : : : ; pbn
1
b
0
b
b
=
ps 1
C xs p1 ; : : : ; p n
@ps
ebs pb1 ; : : : ; pbn
= 0:
(64)
Bertrand, Cournot and Monopolistically Competitive Equilibria
50
Because of (63), (10) and (64) imply (60) and (61).
Note that, Remarks 13 and 15, (63) and the condition C 00 (x) 0 imply
"
#
!
@
1
ps 1
C 0 xs pb1 ; : : : ; ps ; : : : ; pbn
@ps
ebs pb1 ; : : : ; ps ; : : : ; pbn
2
3
"
#
@ebs (pb1 ;:::;ps ;:::;pbn )
1
@ps
5
= 1
+ ps 4
2
b
b
b
b
b
es p1 ; : : : ; ps ; : : : ; pn
es p1 ; : : : ; ps ; : : : ; pbn
C
00
xs
pb1 ; : : : ; ps ; : : : ; pbn
@xs pb1 ; : : : ; ps ; : : : ; pbn
@ps
> 0:
In addition, it is easy to verify that
#
"
1
pbs 1
ebs pb1 ; : : : ; ps ; : : : ; pbn
C 0 xs pb1 ; : : : ; ps ; : : : ; pbn
is negative when ps is near zero and positive when ps is large. Thus, there exists a
unique solution, ps ; to
@
Rs pb1 ; : : : ; ps ; : : : ; pbn
C xs pb1 ; : : : ; ps ; : : : ; pbn
= 0:
(65)
@ps
In addition, at the price, ps ; for which (65) is satis…ed,
@2
Rs pb1 ; : : : ; ps ; : : : ; pbn
C xs pb1 ; : : : ; ps ; : : : ; pbn
2
@p
" s
"
#
!#
@
1
0
b
b
=
ps 1
C xs p1 ; : : : ; p s ; : : : ; p n
@ps
ebs pb1 ; : : : ; ps ; : : : ; pbn
@xs pb1 ; : : : ; ps ; : : : ; pbn
@ps
"
#
!
@ 2 xs pb1 ; : : : ; ps ; : : : ; pbn
1
0
b
b
+ ps 1
C
x
p
;
:
:
:
;
p
;
:
:
:
;
p
s
s
1
n
@p2s
ebs pb1 ; : : : ; ps ; : : : ; pbn
"
"
#
!#
@xs pb1 ; : : : ; pbn
@
1
0
b
b
=
ps 1
C
x
;
:
:
:
;
p
;
:
:
:
;
p
p
s
s
1
n
@ps
@ps
ebs pb1 ; : : : ; ps ; : : : ; pbn
< 0:
Thus, the price ps at which (65) is satis…ed must be
pbs = argmax Rs pb1 ; : : : ; ps ; : : : ; pbn
ps
C xs pb1 ; : : : ; ps ; : : : ; pbn
:k
Bertrand, Cournot and Monopolistically Competitive Equilibria
7.1.
51
Symmetric Bertrand Equilibrium.
Proposition 10. When condition (2) holds, there exists a symmetric Bertrand equilibrium in which each …rm s charges
pbs = pb
and supplies
xb
x pb = m (1
1
1
pb
)
(1
)
n[ (1
)
][ (1 1
1
)
]
(66)
b
where p is the solution to
pb =
C 0 x pb
h
1
1
C 0 m (1
1
n
1
1
)
+
1
(1
1
pb
(1
=
1
)n
)
n[ (1
i
(67)
1
)
][ (1 1
1
)
]
:
1
"b (n)
1
The Bertrand output xb can also be obtained as the solution to
p x
b
= mn( 1 )
1
x
b
1
=
h
1
=
C 0 xb
1
1
"b (n)
C 0 xb
1
1
1
n
+
1
(1
1
)n
i
1
(68)
:
Proof: When all …rms charge pb ; Remark 6 implies that the demand of each …rm
is as given in (66) and Remark 16 implies that.
"bs (p; : : : ; p) = "b (n) :
Equation (67) is obtained by substituting x pb and
"b (n) =
1
1
1
n
+
1
(1
1
)n
in (60) and (61).
Note that a solution to (67) clearly exists since the demand in (66) grows without
bound as pb approaches zero.
Finally, note that when all …rms supply xb ; pb = p xb where p (x) is the function
de…ned Remark 2. Thus, we can substitute p xb for pb and xb for x pb in equation
(67) to get (68).k
Bertrand, Cournot and Monopolistically Competitive Equilibria
52
Remark 40. The relationship between the Bertrand equilibrium price, pb ; and the
marginal cost in (67) is again the standard result
price = h
marginal cost
1
demand elasticity
1
i:
that as noted in Remarks 28 and 36 also holds in the Cournot and Dixit-Stiglitz
equilibria.
Remark 41. As is true with the symmetric Cournot equilibrium, it will often be
useful to emphasize the dependence of xb and pb on m and n by using the notation
xb (m; n) and pb (m; n) to denote the Cournot equilibrium output and price. When
there is no danger of confusion we will simply denote xb (m; n) and pb (m; n) by xb
and pb :
The Case When Marginal Cost is Constant. In this case, (68) becomes
mn( 1 )
1
1
xb
c
=
1
"b (n)
1
so that
Also (67) becomes
2
3
(
)
cn 1
5
xb = 4
1
m 1 "b (n)
1
c
pb =
1
Finally note that
pb
2
c = c4
1
"b (n)
1
1
1
"b (n)
1
1
:
:
3
15 :
7.2. Uniqueness of the Bertrand Equilibrium. In the previous section we
characterized the symmetric Bertrand Equilibrium in which all …rms charge the same
price. In this section we prove that this is the only Bertrand equilibrium by establishing the following Proposition.
Proposition 11. Assume that condition (2) holds. In a Bertrand equilibrium, all
…rms must charge the same price.
Bertrand, Cournot and Monopolistically Competitive Equilibria
Proof: We can demonstrate that, for
Pn
1
t=1
that solves (61) by demonstrating that, for
1
…xed, there is one value of ps
pt
Pn
t=1
53
1
1
…xed,
pt
C 0 xs pb1 ; : : : ; pbn
1
1
"bs (pb1 ;:::;pbn )
is decreasing in ps . First, we note that the expression (5) for the demand function tells
Pn 1 1
…xed, xs pb1 ; : : : ; pbn is decreasing in ps : Since C 00 (x) 0;
us that, for
t=1 pt
C 0 xs pb1 ; : : : ; pbn
Pn 1 1
is nonincreasing in ps if
is …xed. Also Remark 14 implies that, for
t=1 pt
Pn 1 1
1
…xed, "b pb ;:::;p
; and hence
b
t=1 pt
s( 1
n)
1
1
1
"bs (pb1 ;:::;pbn )
is decreasing in ps : k
7.3.
Price and Supply Related to Market Size and the Number of Firms.
Proposition 12. Assume that condition (2) holds. In that case, xb (m; n) and
pb (m; n) are increasing functions of the market size parameter m. Also, pb (m; n)
is a decreasing function of the number of …rms, n. In addition, there exist an n
^;
b
independent of m; such that, if, n > n
^ , then x (m; n) is a decreasing function of the
number of …rms, n.
Proof: Let’s de…ne
(p; m; n)
C 0 (x (p; m; n))
p
1
1
1
1
n
+
1
(1
1
)n
1
!
(69)
where x (p; m; n) is de…ned in (46). Equation (67) which determines pb (m; n) can be
rewritten as
pb (m; n) ; m; n = 0:
(70)
Implicitly di¤erentiating (70) we get
@pb (m; n)
=
@n
pb (m; n) ; m; n
b
p (p (m; n) ; m; n)
n
(71)
Bertrand, Cournot and Monopolistically Competitive Equilibria
54
where (69) implies that
p
C 00 (x (p; m; n)) @x
@p
(p; m; n) =
(72)
p
0
C (x (p; m; n))
p2
and
@x
C 00 (x (p; m; n)) @n
n (p; m; n) =
p
h
i
1
+
n2
h
1
(1
1
)
1
n
1
+
1
(1
1
)n
i2 :
These expressions together with (2), (47), (48) and (71) imply that
p
(p; m; n) < 0;
n
(p; m; n) < 0
(73)
and
@pb (m; n)
< 0:
@n
Implicitly di¤erentiating (70) once again we get
pb ; m; n
b
p (p ; m; n)
@pb (m; n)
=
@m
m
(74)
where (69) implies that
m
(p; m; n) =
@x
C 00 (x (p; m; n)) @m
p
(75)
and where (2) and (46) imply that
@x
=
@m
1
1
m( 1
1
1)
n(
1
1
)[1 ( 1 )] p
1
1
> 0:
(76)
Equations (75) and (76) imply that
m
(p; m; n) > 0:
(77)
Bertrand, Cournot and Monopolistically Competitive Equilibria
55
Combining (73), (74) and (77) we get
@pb (m; n)
> 0:
@m
For the purpose of computing
Q (x; m; n)
mn( 1 )
1
@xb (m;n)
@m
1
1
@xb (m;n)
;
@n
and
let’s de…ne
!
1
1
1
+
x
(1
)n
1
n
1
1
C 0 (x)
and rewrite (68) as
Q xb (m; n) ; m; n = 0:
(78)
Implicitly di¤erentiating (78) we get
@xb (m; n)
=
@m
and
Qm xb (m; n) ; m; n
Qx (xb (m; n) ; m; n)
(79)
Qn xb ; m; n
Qx (xb ; m; n)
(80)
@xb (m; n)
=
@n
Since
Qx (x; m; n)
= (
1) mn( 1 )
1
1
1
1
n
1
1
+
1
1
)n
(1
!
(81)
2
x
C 00 (x)
< 0
and
Qm (x; m; n)
= n( 1 )
the expression for
is
@xb
@m
1
1
1
1
1
n
+
1
(1
1
1
)n
!
x
1
>0
in (78) is positive. When we compute Qn (x; m; n) ; the result
Qn (x; m; n) = mn( 1 ) 2 x
1
where
1
+
1 1
n
1
1
1
(1
1
1
)
1
1
1
n
1
n
+
+
1
(1
1
(1
1
)n
1
)n
1
2
!
!
:
Bertrand, Cournot and Monopolistically Competitive Equilibria
56
has a negative …rst term and a positive second term. However, when n is su¢ ciently
large the positive second term in this expression is near zero and
'
1
(1
=
1 (1
)
) < 0:
Thus, when n is su¢ ciently large, Qn xb ; m; n is negative and (80) and (81) imply
that
@xb (m; n)
< 0: k
@n
.
7.4. Pro…tability, Market Size and the Number of Firms. Using the results
established in Proposition 12 relating the Bertrand price and per …rm output to the
size of the market and the number of …rms we can now see how increases in the
market’s size and the number of …rms a¤ects per unit pro…ts and total pro…ts.
Corollary 13. Assume that condition (2) holds. The per unit pro…t,
pb (m; n)
AC xb (m; n) ;
and pro…ts per …rm,
pb (m; n) xb (m; n)
C xb (m; n) ;
are both increasing in m; if
xb (m; n) < x (0) :
Proof: Proposition 12 asserts that an increase in m causes both pb (m; n) and
xb (m; n) to increase. When
xb (m; n) < x (0) :
the increase in xb (m; n) causes AC xb (m; n) to fall and Thus, per unit pro…ts,
pb (m; n)
AC xb (m; n) ;
and xb (m; n) both increase with m; and pro…ts per …rm,
pb (m; n) xb (m; n)
C xb (m; n) = pb (m; n)
must also increase with m: k
AC xb (m; n)
xb (m; n) ;
Bertrand, Cournot and Monopolistically Competitive Equilibria
57
Pro…ts of a New Entrant and the Relationship Between the Number
of Firms and Bertrand Pro…ts. We can now prove that the Bertrand pro…ts
are decreasing in n: This result is obtained by showing that an entrant who faces n
…rms each charging pb (m; n) will earn pro…ts that are lower than the n …rm Bertrand
pro…ts
pb (m; n) xb (m; n) C xb (m; n)
but are higher than the pro…ts
pb (m; n + 1) xb (m; n + 1)
C xb (m; n + 1)
earned in the n + 1 …rm Bertrand equilibrium.
We begin by observing that (5) implies that, if we have n …rms producing and
selling at price p~; and a new entrant charges p its sales will be
xn+1 (~
p; : : : ; p~; p) = m (1
1
)
p
1
p
1
+ n~
p
1
[ (1
)
][ (1 1
1
)
]
:
Considered as a function of p; xn+1 (~
p; : : : ; p~; p) is the demand function faced by the
new entrant. Now let’s de…ne the new entrant’s inverse demand function p (x; p~; n)
implicitly by
xn+1 (~
p; : : : ; p~; p (x; p~; n)) = x:
Remark 42. We are going to make use of the fact that the problem of choosing p
to maximize
[Rn+1 (~
p; : : : ; p~; p) C (xn+1 (~
p; : : : ; p~; p))]
can also be solved by choosing x to maximize
[p (x; p~; n) x
C (x)] :
Formally,
max [Rn+1 (~
p; : : : ; p~; p)
p
= max [p (x; p~; n) x
x
C (xn+1 (~
p; : : : ; p~; p))]
C (x)] :
Lemma 14. Assume that condition (2) holds. Then, p (x; p~; n) is an increasing function of p~ and a decreasing function of n: As a consequence, the function
max [p (x; p~; n) x
x
C (x)]
= max [Rn+1 (~
p; : : : ; p~; p)
p
C (xn+1 (~
p; : : : ; p~; p))]
is also an increasing function p~ and a decreasing function of n:
Bertrand, Cournot and Monopolistically Competitive Equilibria
58
Proof: Note that p (x; p~; n) is the solution to
(p (x; p~; n) ; x; p~; n) = 0;
where
(p; x; p~; n)
x
m
1
(1
)
p
1
p
1
+ n~
p
1
[ (1
)
][ (1 1
1
)
]
:
Implicitly di¤erentiating we get
pp~ (x; p~; n) =
~; n) ; x; p~; n)
p~ (p (x; p
p
and
pn (x; p~; n) =
(p (x; p~; n) ; x; p~; n)
:
(p (x; p~; n) ; x; p~; n)
:
~; n) ; x; p~; n)
p (p (x; p
n
Since (2) implies
~; n)
p~ (p; x; p
<0
and since
n
(p; x; p~; n) < 0;
p
(p; x; p~; n) > 0;
and
we have
pn (x; p~; n) < 0
and
pp~ (x; p~; n) > 0:
Since p (x; p~; n) an increasing function p~ and a decreasing function of n for each
x; the same is true of
p (x; p~; n) x C (x) :
This implies that
max [p (x; p~; n) x
x
C (x)]
is an increasing function p~ and a decreasing function of n: k
By applying Lemma 14 twice and using the fact that
pb (m; n + 1) < pb (m; n)
we obtain the following proposition.
Bertrand, Cournot and Monopolistically Competitive Equilibria
59
Proposition 15. Assume that condition (2) holds. The pro…ts earned by entrant
who faces n …rms each charging pb (m; n) exceed the n + 1 …rm Bertrand pro…ts
pb (m; n + 1) xb (m; n + 1)
C xb (m; n + 1)
but are lower than the n …rm Bertrand pro…ts
pb (m; n) xb (m; n)
C xb (m; n) :
Formally,
pb (m; n) xb (m; n)
C xb (m; n)
= max Rn pb (m; n) ; : : : ; pb (m; n) ; p
p
C xn pb (m; n) ; : : : ; pb (m; n) ; p
> max Rn+1 pb (m; n) ; : : : ; pb (m; n) ; p
p
C xn+1 pb (m; n) ; : : : ; pb (m; n) ; p
> max[Rn+1 pb (m; n + 1) ; : : : ; pb (m; n + 1) ; p
p
C xn+1 pb (m; n + 1) ; : : : ; pb (m; n + 1) ; p ]
= pb (m; n + 1) xb (m; n + 1)
C xb (m; n + 1) :
Proof: Since
max [p (x; p~; n) x
x
C (x)]
is a decreasing function of n;
pb (m; n) xb (m; n)
C xb (m; n)
= max p x; pb (m; n) ; n
1 x
x
> max p x; pb (m; n) ; n x
C (x)
C (x)
x
= max Rn+1 pb (m; n) ; : : : ; pb (m; n) ; p
p
C xn+1 pb (m; n) ; : : : ; pb (m; n) ; p
:
Also since
max [p (x; p~; n) x
x
C (x)]
is an increasing function p~ and since Proposition 12 implies that
pb (m; n + 1) < pb (m; n) ;
max Rn+1 pb (m; n) ; : : : ; pb (m; n) ; : : : ; p~; p
p
= max p x; pb (m; n) ; n x
C (x)
x
< max p x; pb (m; n + 1) ; n x
x
b
= p (m; n + 1) xb (m; n + 1)
C xn+1 pb (m; n) ; : : : ; pb (m; n) ; p
C (x)
C xb (m; n + 1) : k
Bertrand, Cournot and Monopolistically Competitive Equilibria
60
When the Market is Large Enough to Ensure Bertrand Pro…ts. Let’s
…x the number of …rms at n: We now show, by an argument analogous to that used
to establish Proposition 5, that we can always make m large enough so that we can
have n …rms producing pro…tably at the Bertrand outcome. For the given n; we let
n(1
mb (n; )
1
C 0 (x ( ))
)
x( )
1
1
1
;
eb (n)
the Bertrand analog of m (n; ) : Since,
eb (n) > ec (n) ;
mb (n; ) > m (n; ) :
Proposition 16. Assume that condition (2) holds. If m = mb (n; ) and there are
n …rms, then the Bertrand equilibrium output is
xb (m; n) = x ( ) ;
the Bertrand price charged by all …rms is
pb (m; n) = mb (n; ) n( 1
1)
x( )
1
=
C 0 (x ( ))
1
1
eb (n)
:
(82)
and this price exceeds average cost so that all …rms earn a pro…t.
If m > mb (n; ) and n is su¢ ciently large, then the Bertrand equilibrium output
is
xb (m; n) x ( ) ;
the Bertrand price charged by all …rms is
b
p (m; n)
C 0 (x ( ))
1
1
eb (n)
and this price exceeds average cost so that all …rms earn a pro…t.
If m = mb (n; ) and there are n0 < n …rms, and n0 is su¢ ciently large then the
Bertrand equilibrium output is
xb (m; n0 )
x( );
Bertrand, Cournot and Monopolistically Competitive Equilibria
61
the Bertrand price charged by all …rms is
C 0 (x ( ))
pb (m; n0 )
1
eb (n)
1
and this price exceeds average cost so that all …rms earn a pro…t.
If m = mb (n; ) and there are n0 > n …rms, and n is su¢ ciently large then the
Bertrand equilibrium output is
xb (m; n0 )
x( );
the Bertrand price charged by all …rms is
C 0 (x ( ))
pb (m; n0 )
1
eb (n)
1
:
Proof: It is immediate to verify that mb (n; ) has been chosen so that x ( )
satis…es condition (68) and
xb (m; n) = x ( ) :
The Bertrand equilibrium price in (82) is obtained by substituting x ( ) for x pb in
(67). Since
C 0 (x ( )) = (1
) AC (x ( ))
(82) implies
pb (m; n) =
(1
) AC (x ( ))
1
1
eb (n)
> AC (x ( )) :
The last inequality follows from Remark 13.
If m > m (n; ) and there are n …rms, the results that the Bertrand equilibrium
output exceeds x ( ) and the Bertrand price, pb (m; n) ; charged by all …rms exceeds
that given in (52) are corollaries of Proposition 12. The fact that all …rms make a
pro…t is a consequence of Corollary 13 if
xb (m; n) < x (0) :
If
xb (m; n)
x (0)
the fact that
pb (m; n) > C 0 xb (m; n)
AC xb (m; n)
implies that all …rms make a pro…t.
The remainder of the proof follows immediately from Proposition 12 and Corollary
13.k
Bertrand, Cournot and Monopolistically Competitive Equilibria
62
8. The Collusive Outcome
De…nition 5. Suppose that there are n …rms and that the market size parameter is
m: In the symmetric collusive outcome, output per …rm, xM (m; n) ; is chosen to
maximize industry pro…ts. Thus,
xM (m; n) = argmax [p (x) x
C (x)]
x
where
p (x) = mn( 1 ) 1 x
1
is the inverse demand function de…ned in Remark 2. The collusive price charged by
each …rm is
pM (m; n) = p xM (m; n)
Remark 43. We could, of course, have proceeded more generally in de…ning the
collusive outcome by not requiring that all …rms produce the same output and charge
the same price. It is easy to see, however, that, because of the symmetry of the
demand function (5), industry pro…t maximization implies that all …rms act alike.
Remark 44. The function
[p (x) x C (x)]
= mn( 1 ) 1 x
C (x)
relating pro…ts per …rm to output per …rm, x; is a strictly concave function.
Proof:
h
i
C (x)
@ mn( 1 ) 1 x
@x
= mn( 1 ) 1 x
1
C 0 (x)
(83)
and
h
@ 2 mn( 1 ) 1 x
@x2
i
C (x)
=
(
1) mn( 1 ) 1 x
1
C 00 (x) < 0: k
(84)
Proposition 17. In the symmetric collusive outcome, each …rm s produces xM where
xM is the solution to
C 0 xM
1
=
mn( 1 ) 1 xM
:
(85)
and charges
pM = p xM = mn( 1 )
1
xM
1
Bertrand, Cournot and Monopolistically Competitive Equilibria
63
The Collusive price pM can also be obtained as the solution to
pM =
C 0 x pM
(86)
:
where, because of Remark 6,
xM = x pM
m (1
1
)
pM
1
(1
)
n[ (1
)
][ (1 1
1
)
]:
Proof: Since [p (x) x C (x)] is strictly concave, xM is characterized by the …rst
order condition obtained by setting the …rst derivative computed in (83) equal to
zero. Equations (85) and (86) follow immediately from that …rst order condition.k
Remark 45. The relationship between the collusive price, pM ; and the marginal cost
in (86) is once more the standard result
price = h
marginal cost
1
1
demand elasticity
i:
that as noted in Remarks 28, 36 and 40 also holds in the Dixit-Stiglitz, Cournot and
Bertrand equilibria.
Remark 46. Although xM (m; n) and pM (m; n) depend on m; n and : we have
already suppressed this dependence in our statement of Proposition 17. In the subsequent discussion, we will simply denote xM (m; n) and pM (m; n) by xM and pM :when
there is no danger of confusion or when there is no need to emphasize the dependence
of these outcomes on m and n.
9.
Bertrand and Cournot Equilibria and the Collusive Outcome
Compared
Proposition 18. At every m and n; the Bertrand equilibrium price, pb (m; n) ; is
lower than the Cournot equilibrium price, pc (m; n) ; which is, in turn, lower than the
collusive price, pM (m; n) : In addition, the Bertrand equilibrium supply, xb (m; n) ; exceeds the Cournot equilibrium supply, xc (m; n) which, in turn, exceeds the Collusive
output, xM (m; n) : Finally, The Bertrand equilibrium pro…ts,
p xb (m; n) xb (m; n)
C xb (m; n) ;
are lower than the Cournot equilibrium pro…ts,
p (xc (m; n)) xc (m; n)
C (xc (m; n))
which are, in turn, lower than the collusive pro…ts
p xM (m; n) xM (m; n)
C xM (m; n) :
Bertrand, Cournot and Monopolistically Competitive Equilibria
64
Proof: Recalling, Remarks 36, 40 and 45, let’s rewrite (36), (67) and (86) as
pc
=h
C 0 (x (pc ))
1
and
pb
=h
C 0 (x (pb ))
1
pM
=h
C 0 (x (pM ))
1
1
1
"c (n)
1
1
"b (n)
i
(87)
i
(88)
1
1
(1
)
1
respectively. Remarks 13, 20 and 24 imply that
"b (n) > "c (n) >
1
1
i:
(89)
:
Thus, (87), (88) and (89) imply
pb
pc
pM
<
<
:
C 0 (x (pb ))
C 0 (x (pc ))
C 0 (x (pM ))
(90)
Since
x0 (p)
is decreasing in p and C 00 (x)
0
C0
p
(x (p))
is increasing in p and (90) implies
pb < p c < p M :
Since x0 (p) < 0; these inequalities imply
x pb > x (pc ) > x pM :
Because of Remark 44 and the fact that
@ p xM xM
@x
C xM
p (x) x
C (x)
the function
= 0;
(91)
Bertrand, Cournot and Monopolistically Competitive Equilibria
65
p(x )x − C (x )
p (x M )x M − C (x M )
( )
( )
p xc xc − C xc
p (x b )x b − C (x b )
xM
xc
xb
x
is as shown in Figure 5.
As Figure 5 and Remark 44 imply,
p (x) x
C (x)
is a decreasing function above xM ; (91) implies
p xM xM
C xM > p (xc ) xc
C (xc ) > p xb xb
C xb : k
Remark 47. Proposition 18, of course, implies that the Bertrand pro…ts are negative whenever the Cournot pro…ts are negative and the Cournot pro…ts are positive
whenever the Bertrand pro…ts are positive. As a consequence we have the following
corollary of Propositions 6 and 18.
Corollary 19. For every m; there exists an n such that when the market size is m
and there are n n …rms, all …rms lose money in the Bertrand equilibrium.
10. Bertrand-Free Entry Equilibrium
As in the discussion of the Cournot free-entry equilibrium we assume that there is an
in…nity of potential …rms all of which possess the same cost function satisfying the
conditions of either Cost Case 1 or Cost Case 2.
Bertrand, Cournot and Monopolistically Competitive Equilibria
66
De…nition 6. In a Bertrand free-entry equilibrium there are nb …rms that produce. The vector of prices charged by these …rms, pb1 ; : : : ; pbnb ; is a Bertrand equilibrium and each of these producing …rms makes a nonnegative (but possibly zero)
pro…t. Thus, for each s;
maxRs pb1 ; : : : ; ps ; : : : ; pbnb
ps
C xs pb1 ; : : : ; ps ; : : : ; pbnb
= Rs pb1 ; : : : ; pbnb
C xs pb1 ; : : : ; pbnb
is nonnegative. In addition, none of the non-producing …rms can make a pro…t. Thus,
the …nal condition satis…ed in a Bertrand free-entry equilibrium is that
maxRs pb1 ; : : : ; pbnb ; p
p
C xn+1 pb1 ; : : : ; ps ; : : : ; pbnb ; p
is nonpositive.
In discussing the Bertrand free-entry equilibria, we make use of the fact established
in Proposition 11 that, for each n; the unique Bertrand equilibrium is the symmetric
Bertrand equilibrium shown to exist and described in Proposition 10. Of course, in
that symmetric equilibrium, all …rms charge the same price pb (m; n) :
It appears that, in general, a Bertrand free-entry equilibrium may fail to exist.
In any case, because of Proposition 15, it is not possible to use an argument that
parallels the one used to establish the existence of a Cournot free-entry equilibrium.
We can however, make some observations.
Remark 48. Note that if we have a Bertrand free-entry equilibrium with nb …rms,
then there will also be a Cournot free-entry equilibrium with nc …rms where
nc
nb :
This is true since as we noted in Remark 47, the Cournot pro…ts are positive whenever
the Bertrand pro…ts are positive .
Proposition 20. If m
maxmb (n; ) ; then there exists an N > n
~ such that all
n n
~
…rms earn a nonnegative pro…t in the Bertrand equilibrium with n …rms when
n
N
Bertrand, Cournot and Monopolistically Competitive Equilibria
67
and all …rms su¤er a loss in the Bertrand equilibrium with n …rms when
n > N:
If, in addition, n
~ is su¢ ciently large,
xb (m; N ) < x ( ) :
If there exists a Bertrand-free entry equilibrium then there are N …rms in equilibrium.
Proof: From Proposition 16, we know note that if m
maxmb (n; ) ; then …rms
n n
~
make a pro…t in the n …rm Bertrand equilibrium when n
n
~ . From Corollary 19
we know that for every m; there is some n; such that pro…ts are negative if there are
more than n …rms. There must, therefore, exist some N 2 [~
n; n) for which all …rms
earn a nonnegative pro…t in the Bertrand equilibrium with n …rms when
n
N
and all …rms su¤er a loss in the Bertrand equilibrium with n …rms when
n > N:
Proposition 16 also implies that
xb (m; N ) < x ( )
if n
~ and, therefore N; is su¢ ciently large.
Since …rms make a nonnegative pro…t in the n …rm Bertrand equilibrium when
n
N;
Proposition 15 implies that entry will be pro…table when there are less than N …rms
producing in the Bertrand equilibrium. Thus, we can’t have a Bertrand-free entry
equilibrium with less than N …rms. We can’t have a Bertrand-free entry equilibrium
with more than N …rms because …rms must su¤er a loss in the Bertrand equilibrium
when there are more than N …rms. k
Remark 49. There will exist a Bertrand-free entry equilibrium if
max RN +1 pb (m; N ) ; : : : ; pb (m; N ) ; p
p
C xN +1 pb (m; N ) ; : : : ; pb (m; N ) ; p
< 0:
It appears that this condition can hold for some m: It also seems clear that if this
condition holds for some m; the inequality can be reversed by simply raising m slightly.
Bertrand, Cournot and Monopolistically Competitive Equilibria
68
Although we don’t know whether a Bertrand-Free Entry Equilibrium always exists
we do know that if one does exist in a large market, it will approximate the DixitStiglitz equilibrium.
Proposition 21. When the market size parameter m is su¢ ciently large, the amount
produced by each of the nb …rms in a Bertrand free-entry equilibrium, if one exists,
is
xb m; nb ' x ( )
and the price charged by each of these …rms is
pb m; nb ' AC (x ( )) :
The proof of this proposition is analogous to that given for Proposition 9.
11. A Case When Our Elasticity Condition Fails
We will not explicitly consider the case in which
1
<
1
1
but we will consider the borderline case in which
1
=
1
1
:
In this case(1) reduces to
ps (x1 ; : : : ; xn ) = mxs
and (5) reduces to
xs (p1 ; : : : ; pn ) = m (1
Remark 50. When
1
=
1
1
1
1
)
ps :
;
the Cournot and Bertrand equilibria and the collusive outome all coincide. In each
of these equilibria all …rms charge
p
M
= ps x
M
=m x
and supply the amount xM that solves (92).
M
C 0 xM
=
;
(1
)
(92)
Bertrand, Cournot and Monopolistically Competitive Equilibria
69
In the case of constant marginal cost, the solution to (92) is
x
M
m (1
=
c
)
1
:
So all …rms produce that amount and charge
pM =
c
(1
)
:
which, as noted in Remark 30, is the Dixit-Stiglitz price p ( ) :
The equilibrium price and output are independent of the number of …rms, n;
but, when marginal cost is increasing, both increase with the size of the market as
measured by m: When marginal cost is constant, the equilibrium price remains at
p ( ) whether the market is large or small, but xM increases with the size of the
market. The equilibrium coincides with the Dixit-Stiglitz equilibrium if m = m ( )
where
C 0 (x ( ))
m( ) =
:
x ( ) (1
)
In that case, …rms obviously earn no pro…ts.
When m < m ( ) ;
xM < x ( ) ;
and …rms earn negative pro…ts. When m < m ( ) ; and marginal cost is increasing,
we also have
C 0 xM
pM =
< p( ):
(1
)
The case in which m < m ( ) and marginal cost is increasing is illustrated Figure 6.
In that case,
C 0 xM
AC xM > ps xM =
:
(1
)
Figure 7 illustrates a case in which marginal cost is constant and m < m ( ) so
that
1
m (1
)
1
F
= xM < x ( ) =
1
:
c
c
In Figure 7, as in Figure 6, we also have
AC xM > pM =
c
(1
)
:
Bertrand, Cournot and Monopolistically Competitive Equilibria
MC
AC
AC (x M )
AC (x(α )) =
MC(x(α ))
1−α
( )
( )
MC x M
1−α
MC (x(α ))
p xM =
( )
Demand
MC x M
xM
MR
x(α )
x(0 )
AC (x )
( ) 1 −cα
M
p xM =
AC
c
Demand
MR
xM
x(α )
70
Bertrand, Cournot and Monopolistically Competitive Equilibria
71
When m > m ( ) ;
xM > x ( ) ;
and …rms earn positive pro…ts. When m > m ( ) ; and marginal cost is increasing,
we also have
C 0 xM
> p( ):
p xM =
(1
)
Finally, equilibrium pro…ts,
m xM
1
C xM
are independent of n: Thus, if pro…ts are positive because m > m ( ) ; the entry of
new …rms has no impact on pro…ts. For this reason, neither the Cournot nor Bertrand
free-entry equilibria exist in this case. If m = m ( ) ; there exist both Cournot and
Bertrand free-entry equilibrium which coincides with the Dixit-Stiglitz equilibrium
and any number of …rms might produce in equilibrium. When m < m ( ) ; there
exist trivial Cournot and Bertrand free-entry equilibrium in which no …rms produce.
Proof: When
1
=
1
;
1
choosing xs to maximize
Rs (xc1 ; : : : ; xs ; : : : ; xcn )
C (xs ) = mx1s
= mxs
C (xs )
C (xs )
is equivalent to choosing ps to maximize
C xs pb1 ; : : : ; ps ; : : : ; pbn
Rs pb1 ; : : : ; ps ; : : : ; pbn
= m (1
1
1
)
ps
C m (1
1
1
)
ps
:
Thus, in the Cournot, Bertrand and Collusive cases, …rms solve the same problem
and xc = xb = xM : Because < 1; and C 00 (xs ) > 0;
Rs (xc1 ; : : : ; xs ; : : : ; xcn )
C (xs ) = mx1s
C (xs )
is clearly a strictly concave function of xs : Thus, xc = xb = xM is still characterized
by (29) which now becomes (92), a single equation in the one unknown xc : Note that,
since the solution to (92) is unique and the same for all s, the only equilibrium is one
in which all …rms s supply the same amount. Note also that, because the equation
that determines xM is independent of n; the solution xM is also independent of n: It is,
Bertrand, Cournot and Monopolistically Competitive Equilibria
72
furthermore, easy to check by implicitly di¤erentiating (92) that xM is an increasing
function of m:
The fact that …rms earn negative pro…ts when
xM < x ( )
follows from the fact
AC xM
M C (xM )
rises as xM falls below x ( ) ; while
pM
M C (xM )
remains constant at 1 1 as xM falls below x ( ) :
Conversely, as xM rises above x ( ) ;
AC xM
M C (xM )
falls while
remains constant at
pM
M C (xM )
1
1
: Thus, …rms earn positive pro…ts when
xM > x ( ) : k
12. Summary:
For the special case of the inverse demand functions given in (1) and the demand
functions given in (5) we have described the Cournot and Bertrand equilibria for a
…xed number of …rms. Our characterizations of these equilibria have made it possible to determine how they are a¤ected by the size of the market and the number of
…rms. For the special cases we considered, we were also able to prove the existence of
Cournot free-entry equilibria in which the number of …rms is determined endogenously
In addition, we were able to prove that, in a large market, the Cournot free-entry equilibria approximate the Dixit-Stiglitz monopolistically competitive equilibria. While
we were unable to establish a general existence result for Bertrand free-entry equilibria, we were able to prove that, when these equilibria exist, they are unique and that
in large markets they also approximate the Dixit-Stiglitz equilibria.
While the results that emerged were primarily as expected, the arguments required
were more involved than those normally required in the case when …rms sell goods
Bertrand, Cournot and Monopolistically Competitive Equilibria
73
that are perfect substitutes. It is natural to ask to what extent these arguments can
be extended. In searching for extensions, it would seem to be necessary to restrict
attention to cases in which an analog of our elasticity condition continues to hold.
One such extension that appears promising is that in which the demand functions
are derived from utility maximization of the utility function
"
!#
n
X
u (y; x1 ; : : : ; xn ) = y + mh
xt1
t=1
where h0 ( ) > 0; h00 ( ) < 0 and
<
zh00 (z)
< 1:
h0 (z)
We have considered the case of
h (z) =
for which
(93)
z
zh00 (z)
= (1
h0 ( )
):
The extension we propose here would be one in which
zh00 (z)
h0 (z)
1
would become the monopoly elasticity of demand and (93) would be the extension of
(2). In this extension, the monopoly elasticity of demand would, of course, no longer
be constant.
References
[1] Chamberlin, E. (1933) The Theory of Monopolistic Competition. Cambridge, MA:
Harvard University Press.
[2] Dixit, A. and J. Stiglitz (1977) ”Monopolistic Competition and Optimum Product
Diversity,”American Economic Review: 297-308.
[3] Novshek, W. (1980) ”On the Existence of Cournot Equilibrium,”Review of Economic Studies: 85-98.