Equilibria in a Capacity-Constrained Di¤erentiated
Duopoly
Maxim Sinitsyn
McGill University
Preliminary and incomplete.
Abstract
In this paper I analyze the model of price-setting duopoly with capacity-constrained
…rms producing di¤erentiated products. I show that a pure strategy equilibrium exists if the capacities of the …rms are both small or both large. For the other capacity
values, mixed strategy equilibria exist, with the support of the price distribution either being an interval or consisting of a …nite number of points. I show that if before
the price competition the …rms simultaneously choose their capacity levels, their optimal choice of capacities would lead to a pure strategy equilibrium in prices. The
optimal capacity levels decrease with an increase in consumer heterogeneity.
1
Introduction
Recent advances in estimations of demands for di¤erentiated products necessitate research
in the supply side of the market – how the …rms act when facing such demands. While
this question has been studied thoroughly for the …rms that are not constrained in their
production, the literature on the behavior of the capacity-constrained …rms producing
1
di¤erentiated products is scarce. In this work, I characterize and compute the equilibria
that occur in the situations, when the …rms are capacity-constrained.
The literature on price competition with capacity constraints initially considered only
homogeneous products (Bertrand-Edgeworth competition). Due to discontinuities in demand and pro…t functions, pure strategy equilibria do not always exist, thus, it was necessary to work with mixed strategy equilibria. First, the equilibria in symmetric BertrandEdgeworth models were computed for the proportional (Beckmann (1965)) and surplusmaximizing (Levitan and Shubik (1972)) rationing rules for unsatis…ed demand. Kreps
and Scheinkman (1983) characterized the mixed strategy equilibrium for asymmetric capacities using the surplus-maximizing rule. This allowed them to solve a two-stage game,
in which the …rms …rst choose capacities and then compete in prices. They found that the
…rms’choice of capacities in the …rst stage coincides with the Cournot equilibrium – the
production levels the …rms would choose if they were to compete in quantities.
Davidson and Deneckere (1986) argued that this result crucially depends on the assumption of the surplus-maximizing rationing rule, and it does not hold for almost all
other rationing rules. They showed that for the proportional rationing rule, the …rms
necessarily will choose capacities that exceed the Cournot level, and the equilibria are
asymmetric if the capacity cost is small. Furthermore, the mixed strategy equilibria for the
general form of the demand function were characterized by Osborne and Pitchik (1986) for
the surplus-maximizing rationing rule and Allen and Hellwig (1993) for the proportional
rationing rule.
Only recently, models of price competition with capacity constraints were examined for
di¤erentiated products (Bertrand-Edgeworth-Chamberlin competition). Benassy (1989)
proved that a pure strategy equilibrium in these models fails to exist if the degree of
substitutability of the products is large enough.1 In a related paper, Canoy (1996) provided
a parametrized duopoly example in which a pure strategy equilibrium does not exist if
1
Benassy (1989) also shows that for a given degree of product substitutability a pure strategy equilibrium
exists if the number of competitors is large enough.
2
the products are su¢ ciently similar. Thus, it was established that when the products
are su¢ ciently homogeneous only the mixed strategy equilibrium exists. However, no
attempt has been made to characterize and study the equilibrium in mixed strategies for
the Bertrand-Edgeworth-Chamberlin competition. Undertaking this study is the goal of
this paper.
I take the demand to have a logit speci…cation, which is a form of demand for di¤erentiated products, widely used in empirical literature. The …rms produce at a constant
marginal cost, but are limited in production by their capacity constraints. I show that
a pure strategy price equilibrium exists whenever the …rms’ capacities are both small or
both large. For the other values of capacities, a pure strategy price equilibrium does not
exist, and I compute the mixed strategy equilibrium. For the case of symmetric capacities
in the intermediate range, this equilibrium involves a …nite support for the optimal price
distributions. For the case of asymmetric capacities, there are also mixed equilibria, for
which the support of the price distributions is an interval.
The knowledge of the …rms’pro…ts in the pricing game allows me to …nd the equilibrium
in a two-stage game, where the …rms …rst choose the capacity levels. The optimal capacity
levels lead to pure strategy pricing in the second stage. The capacities increase with a
decline in consumer heterogeneity.
2
Basic Model
Consider a market where a particular good is produced by 2 …rms at a zero marginal
cost. Firms compete in prices p. The set of consumers has measure 1: Consumers have
heterogeneous tastes for the goods produced by the …rms. Each consumer receives utility
Ui =
pi + "i from purchasing a product from …rm i. "i is iid standard double exponential.
is the measure of consumer heterogeneity. An outside option gives the consumers a
utility of U0 =
V0 + "0 . Thus, an outside option could be considered as another product
which is sold at a …xed price of V0 . Consumers purchase the product that gives them the
3
highest utility. Then, the demand faced by …rm i is a standard logit demand: Gi (pi ; pj ) =
e
pi =
e
+e
pi =
pj =
+e
V0 =
.
On the supply side, assume that both …rms have zero marginal cost, but can produce
only up to capacity Ki . Given the nature of the consumers’utility functions the rationing
rule for unsatis…ed demand is proportional. If the …rst …rm’s demand G1 (p1 ; p2 ) is greater
than K1 , the remaining (1
demand is (1
K1 ) e
K1 ) consumers have iid draws for "0 and "2 , thus, the residual
e
p2 =
p2 =
+e
V0 =
ei (pi ; pj ) =
G
. In summary, the contingent demand of …rm i is
if Gj (pj ; pi ) 6 Kj
Gi (pi ; pj )
(1
Kj ) e
e
pi =
pi =
+e
V0 =
if Gj (pj ; pi ) > Kj
This section deals with the symmetric case when both …rms have the same capacity
constraint K = K1 = K2 , and the next section will address the case K1 6= K2 . First, I
will examine for what values of K and
a pure strategy equilibrium exists. There are
two di¤erent cases: (a) both …rms produce at the capacity constraint; and (b) both …rms
produce at the level below the capacity constraint.
Production at the capacity constraint.
A single-price equilibrium with both …rms operating at capacity constraints exists when
the parameters satisfy the following conditions2 :
<
1 K
1 2K
V0
+ ln
K
1 2K
, if
1 K
+ ln
1 2K
if
> 0,
K
1 2K
1 K
+ ln
1 2K
(1)
>0
K
1 2K
60
The prices charged by the …rms are
p = V0
2
ln
All derivations are in the Appendix.
4
K
1 2K
(2)
1 K
1 2K
ln
K
1 2K
K
1 2K
+ ln
= 0 for K = Ka
0:178855. For the smaller values of K;
1 K
1 2K
+
is less than zero. Therefore, a single price equilibrium exists for all levels of
consumer heterogeneity when the capacity constraints are small (K is less than Ka ). For
the larger values of K, when the consumer heterogeneity is large enough, a single price
equilibrium with both …rms reaching their capacity levels fails to exist. When V0 is equal
to zero or is negative and K > Ka , this type of a single price equilibrium never exists.
Region 1a in …gure 1 shows the parameter values for which (1) holds.
Figure 1: Types of equilibrium in the symmetric case for V0 = 1.
1
0.9
0.8
0.7
µ
1b
1a
0.6
0.5
0.4
0.3
2
0.2
0.1
5
0
0
0.1
0.2
0.3
0.4
0.5
4
0.6
3
0.7
0.8
K
Production at the level below the capacity constraint.
5
0.9
1
If both …rms are producing at the level below their capacity constraint, they charge the
prices p
that satisfy the following equation:
p
e
=p
V0
+e
The …rm earns a pro…t of
For these prices to be a global maximum,
(3)
V0
p
2e
+e
= p Gi (p ; p ).
must be higher than the monopoly pro…t
on residual demand. Another potential local maximum could occur at a higher price pb,
when the rival is capacity-constrained. This price solves the equation
and gives a pro…t b = pb(1
= pb
e
e
K)
e
p
b
p
b
+e
V0
V0
e
p
b
. p
(4)
;
V0
+e
> b or if
is a global maximum only if
the rival is not capacity-constrained when the price is pb. That is, the following inequalities
must hold:
> b or Gi (p ; pb) 6 K;
(5)
is the solution to (3), and pb is the solution to (4). The region where both
where p
…rms are producing at the level below their capacity constraint is depicted in Figure 1 as
1b.
As
goes to in…nity, the boundary of the region 1b converges to the value Kb . It
p
is possible to …nd the value of Kb . First, (3) could be rewritten as 1 =
When
as 1 =
! 1, e
pb
e
p
b
e
V0
V0
+e
V0
! 1. So,
, from where
p
p
e
p
2e
K ! Kb
+e
0:1858 as
e
V0
=
! 1.
p
pb
! t
! q
e
e
p
b
p
b
+e
V0
+e
V0
V0
.
1:2268. Similarly, (4) could be rewritten
1:2785. The boundary of the region 1b is
= b or pb(1
pb
+e
p
2e
described by the equation
K=1
p
e
K)
e
p
b
p
b
+e
p
V0
e
= p
p
2e
+e
V0
, from where
. Using the values for t and q found previously,
6
The following statements summarize the …ndings about the existence of a pure strategy
equilibrium:
1) For the low levels of the capacity constraint (K 6 Ka
0:178855) a pure strategy
equilibrium always exist with both …rms producing at the capacity constraint level and
charging price p from (2).
2) For the very small region of low capacities (Ka < K 6 Kb
0:1858) a pure strategy
equilibrium exists for the low enough levels of consumer heterogeneity (region 1a in Figure
1). For the high levels of
a pure strategy equilibrium does not exist.
2) For the intermediate levels of the capacity constraint (Kb < K < 0:5) a pure strategy
equilibrium exists for the high levels of consumer heterogeneity
(region 1b in Figure 1)
and for the low levels of consumer heterogeneity (region 1a in Figure 1). There always exist
intermediate values of , for which a pure strategy equilibrium does not exist.
3) For the high levels of the capacity constraint (K > 0:5) a pure strategy equilibrium
exists only when the consumer heterogeneity is high enough (region 1b in Figure 1).
4) For any level of the consumer heterogeneity a pure strategy equilibrium exists for the
low levels of capacity constraint (region 1a in Figure 1) and for the high levels of capacity
constraint (region 1b in Figure 1), but not for the intermediate values. As
decreases, the
region, where a pure strategy equilibrium does not exist, increases.
These conclusions are robust to the changes in V0 . If V0 is increasing, region 1a increases
in size with its boundary approaching the vertical line at 0:5 while region 1b decreases. If
V0 is approaching zero, region 1a decreases in size with its boundary approaching zero for
the values of K > Ka , while region 1b increases.
In the area, where a pure strategy equilibrium does not exist, a mixed strategy equilibrium exists.3 Theorem 1 in Sinitsyn (2006) shows that if the demands are analytic, a
mixed strategy price equilibrium has to have a …nite support. In our case, the demands are
not analytic – they have a kink at the price which makes the rival capacity-constrained.
3
The existence result is due to Glicksberg (1952).
7
Nevertheless, I found that for the symmetric case a mixed strategy equilibrium also has a
…nite support.
Figure 2 illustrates the pricing strategies of the …rms for the small capacity levels.
Figure 2: Equilibrium pricing strategies and pro…ts for K = 0:4.
Prices
1
p
2
0.9
p
p
1
p
1
3
p
4
p
5
0.8
0.5
1
γ
0.4
0.3
µ
Probabilities
0.2
0.1
0
γ1
1
0.5
γ
γ
2
0
0.5
0.4
γ γ
4
3
0.4
0.3
µ
Prof it
0.4
0.3
µ
5
0.2
0.1
0
0.2
0.1
0
π
0.3
0.2
0.1
0
0.5
I …x K to be 0:4 and change the level of consumer heterogeneity
relatively high level of
(
. I start with a
= 0:6) and then decrease it. For the high values of
a
pure strategy price equilibrium exists (the parameters fall in region 1b from Figure 1)
with both …rms operating under their capacity constraints. As
declines, the demands
become more elastic, the equilibrium more competitive, and the optimal prices decrease.
Finally, the prices reach the level p, where it becomes equally pro…table for the …rms to
deviate to the high price p as (p; p) =
(p; p) ( is approximately equal to 0:5). This
8
point is the boundary of the region 1b from Figure 1. If
keeps decreasing, a pure strategy
equilibrium does not exist anymore as the …rms prefer deviation to the high price. However,
an equilibrium with two prices appears.
Each …rm charges two prices – p1 and p2 with corresponding probabilities
1
and
2.
From …gure 2 it is evident that when the two prices are charged the lower of two prices –
p1 –does not decrease as steeply as in the region with one price only. This means that if
p1 were the only price charged, not only there would be a pro…table deviation to the higher
price p2 , but also there would be an incentive to undercut p1 . This does not happen because
the presence of a higher price p2 charged with some probability
2
reduces the undercutting
incentives. Similarly, a price p2 charged with probability 1 will also be undercut, but the
fact that there is a lower price p1 charged with some probability
1
keeps the high price
at p2 . This happens because in the neighborhood of p2 the rival charging a lower price is
always capacity-constrained, so there is an incentive to charge a price higher than p2 . In
equilibrium, this incentive is exactly o¤set by the undercutting incentive and the optimal
price is p2 .
As
keeps decreasing, the incentive to undercut also increase. This puts a greater
weight on p2 (
2
increases) and decreases both prices. Surprisingly, the pro…t increases
slightly as the e¤ect of putting a greater weight on a higher price p2 outweighs the e¤ect of
declining prices. Finally,
declines to such a level (approximately 0:345 in Figure 2) that
again a pro…table deviation appears. The …rms start charging three prices and the cycle
keeps repeating.
In Figure 2 I calculated the mixed strategy equilibria with up to …ve prices charged by
the …rms. It could be observed that the range of the prices charged decreases and …nally
collapses to a point when
reaches approximately 0:27. This is a boundary point of the
region 1a from Figure 1 –so both …rms operate at the capacity constraint and the single
price equilibrium reappears. As could be seen from (2), the optimal price p will increase
with decline of
only if ln
K
1 2K
is positive, which holds for K > 1=3. Since in Figure 2
K = 0:4, the optimal price and the pro…t increase as
9
declines.
Figure 3 illustrates the optimal pricing strategies for the large capacity levels (K is
taken to be 0:75 for this …gure).
Figure 3: Equilibrium pricing strategies and pro…ts for K = 0:75:
Prices
1
p
p
p
2
1
p
3
p5
4
0.5
0
0.5
1
0.45
0.4
0.35
0.3
0.25
0.2
µ
Probabilities
0.15
0.1
0.05
0
γ
1
0.5
γ
0
0.5
0.45
0.4
0.35
0.3
0.25
µ
Profit
0.2
2
0.15
γ
3
0.1
γ
4
γ
5
0.05
0
0.4
π
0.3
π
0
0.2
0.1
0
0.5
0.45
0.4
0.35
0.3
0.25
µ
0.2
0.15
0.1
0.05
0
The movement of prices and the emergence of new equilibria with multiple prices is
similar to the process described for the small capacity levels and illustrated in Figure 2.
The only major di¤erence between the cases with small (K < 0:5) and large capacities is
that for the large capacities, the equilibria with multiple prices do not converge to a single
price equilibrium as
decreases. Instead they approach a mixed strategy equilibrium with
an interval for a price support. In Figure 3 the price range of the mixed strategy equilibrium
for
= 0 is illustrated by the segment [p; p] and the corresponding pro…t is labeled
0.
Figure 1 shows the regions, where mixed strategy equilibria with the …rms charging
multiple prices exist. Each region is numbered in accordance with the number of prices the
10
…rms use. In the unshaded region in Figure 1 both …rms use more than …ve prices in the
equilibrium.
3
Asymmetric Capacities
Now, I will consider the case K1 6= K2 . This will serve as a building block for solving the
two-stage game, in which the …rms …rst choose capacities and then compete in prices.
Pure strategy equilibria
The case where both …rms produce at their capacity level or both …rms produce at the
level below their capacity is handled exactly the same way as in the symmetric case. Figure
4 shows the regions where such equilibria exist as 1a and 1b, correspondingly.
Figure 4: Types of equilibria in the case of asymmetric capacities for V0 = 1 and
1
0.9
(2; 1)
1b
0.8
0.7
(3; 2)
(2; 2)
0.6
K
(1; 2)
1
0.5
(3; 3)
(2; 3)
0.4
0.3
1a
0.2
0.1
0
0
0.2
0.4
0.6
K
11
2
0.8
1
= 0:2.
It seems intuitively plausible that there could exist pure strategy equilibria with one
(larger) …rm producing below its capacity level and another (smaller) …rm producing at its
capacity level. However, the following proposition proves that such equilibria do not exist.
Proposition 1 A pure strategy equilibrium with one …rm producing at the level below its
capacity constraint, and another …rm producing at its capacity constraint does not exist.
Proof. Without loss of generality assume that the …rst …rm produces at the level below
e1 (p1 ; p2 ) < K1 ) and the second …rm produces at its capacity
its capacity constraint (G
e2 (p2 ; p1 ) = K2 ). I will examine the behavior of the pro…t function of the
constraint (G
…rst …rm at p1 . If the …rst …rm charges a price lower than p1 the second …rm operates at
the level below its capacity constraint, otherwise the second …rm could pro…tably increase
its price. Therefore, to the left of p1 the pro…t function of the …rst …rm is
(p1 ; p2 ) =
p1 G1 (p1 ; p2 ). When the …rst …rm charges a price above p1 the second …rm is capacityconstrained, thus the pro…t to the right of p1 is
b1 (p1 ), where G
b1 (p1 ) =
K2 )p1 G
(1
pro…t function at p1 is equal to
@
+ (p1 ; p2 )
p1 =
e
. The
e p1 = +e V0 =
(p1 ;p2 )
= G1 (p1 ; p2 )
@p1
@
+ (p1 ;p2 )
@p1
b
K2 ) G1 (p1 )(1
@
b 1 (p1 ))
G
(p1 ;p2 )
@p1
=
. Using the fact that (1
p1 G1 (p1 ;p2 )(G1 (p1 ;p2 )
b 1 (p1 ))
G
K2 )p1 e
e
p1 =
p1 =
+e
V0 =
=
left hand side derivative of the
p1 G1 (p1 ;p2 )(1
hand side derivative of the pro…t function at p2 is equal to
p1 (1
= (1
@
+ (p1 ;p2 )
@p1
G1 (p1 ;p2 ))
= (1
. The right
b1 (p1 )
K2 )G
b1 (p1 ) = G1 (p1 ; p2 ), we obtain
K2 )G
> 0. Thus, the right hand side derivative
of the …rst …rm’s pro…t function at p1 is greater than the left hand side derivative. The
left hand side derivative has to be greater or equal to zero at p1 (otherwise, there exists a
maximum to the left of p1 ), thus, the right hand side derivative has to be strictly greater
than zero. This means that the pro…t function of the …rst …rm increases to the right of p1 ,
so p1 can not be the maximum.
Therefore, pure strategy equilibria exist only in regions 1a and 1b in Figure 4. For all
other values of K1 and K2 it is necessary to search for the mixed strategy equilibria.
Mixed Strategy Equilibria
12
Figure 4 shows several regions, in which mixed equilibria exist. The numbers in brackets
show the number of prices charged by each …rm, i.e. (3; 2) indicates that the …rst …rm
charges 3 prices, and the second …rm charges 2 prices. The large spaces in the upper-left
and bottom-right correspond to the mixed strategy equilibrium, in which the support of
the price distributions is an interval. A complete characterization of these equilibria and
precise boundaries of the region, where they exist, are to be added.
4
Choice of Capacities (preliminary)
Now, consider a two-stage game, in which the …rms …rst costlessly accumulate capacities,
and then compete in prices. Figure 5 illustrates how the choice of the capacity of the second
…rm a¤ects its pro…ts given that the …rst …rm’s capacity is …xed.
It could be seen from Figure 5 that the optimal choice of capacity for the second …rm
…rst declines as K1 decreases, but always stays smaller than K1 . It keeps declining until
K1 = K2 = K that solves
1 K
1 2K
+ ln
K
1 2K
=
V0
. Afterwards, the optimal value of K2
increases. Thus, there is a unique symmetric equilibrium in a two-stage game. First, both
…rms choose the capacity level K that solves
charge prices p from (2).
To be added.
5
Conclusion
To be added.
13
1 K
1 2K
+ ln
K
1 2K
=
V0
. Then, they both
Figure 5: Pro…ts of the second …rm for the di¤erent capacity levels of the …rst …rm.
0.5
K =0.15
1
K =0.3
0.4
1
K =0.45
1
0.3
K =0.6
1
0.2
K =0.75
1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
K
0.6
0.7
0.8
0.9
1
2
Appendix
Production at the capacity constraint.
First, I will …nd price p , at which the capacity constraint is reached. Gi (p ; p ) = K,
so
2e
e
p =
p =
+e
V0 =
= K. Then, e
p =
(1
2K) = Ke
V0 =
V0
, from where p =
ln
Ke
1 2K
,
and (2) follows.
Now I need to establish conditions under which p is the global maximum. If the …rm
charges any price p below p it will be capacity-constrained, so its pro…t at p will be below
the one at p . In order to check that no prices above p result in a higher pro…t it is enough
to check that the right-hand side derivative of the pro…t function
14
i (pi ; p
ei (pi ; p )
) = pi G
is negative at p .4
e
@ i (p ; p )
ei (p ; p ) + p @ Gi (p ; p )
=G
@pi
@pi
ei (pi ; p ) = (1
When pi is greater than p , G
K) e
e
pi =
pi =
+e
example, Anderson and de Palma (1992), Lemma 1) that if
@ i
@pi
ability given by the multinomial logit, then
(1
K) e
e
pi =
pi =
+e
V0 =
Substituting
, so
e
e
pi =
pi =
+e
=
V0 =
ei (p ; p )
@G
= (1
@pi
e i (p ;p )
@G
@pi
K)
K
.
1 K
;p )
@pi
i (p
i)
Thus,
K
1 K
1
K
1 K
. It is known (see, for
V0 =
i
=
e
e
pi =
pi =
+e
V0 =
is the prob-
ei (p ; p ) =
. In our case, K = G
=
K(1
(1
2K)
K)
(7)
from (7) and p from (2) into (6), we get
@ i (p ; p )
= K + ln
@pi
@
i (1
=
(6)
is less than 0 when 1 + 11
2K
K
ln
V0
Ke
1 2K
K
1 2K
!
K(1
(1
2K)
K)
V0 (1 2K)
(1 K)
< 0 or
(8)
V0
>
1 2K
1 K
K
1 2K
+ ln
.
This is equivalent to (1).
Production at the level below the capacity constraint.
@
ei (pi ; pj ) = Gi (pi ; pj ).
If both …rms operate below their capacity constraints, then G
i (pi ;pj )
@pi
= Gi (pi ; pj )
= p (1
pi
@Gi (pi ;pj )
@pi
= Gi (pi ; pj )
pi Gi (pi ;pj )(1 Gi (pi ;pj ))
@
.
;p
@pi
i (p
)
= 0, so
Gi (p ; p )), which is equivalent to (3).
Firm i can choose to charge a higher price pi for which its rival is capacity-constrained
(Gj (p ; pi ) > K). For such price the pro…t is
i (pi ) = (1
e
K)
e
pi
pi
+e
V0
same argument as in the paragraph above, the optimal price pb should solve
4
i (pi ; pj )
: Following the
= pb
e
e
pi
V0
+e
V0
.
e i (pi ; pj ), where G
e i (pi ; pj ) is logit has a unique maximum, thus once it starts decreasing,
= pi G
it decreases forever.
15
References
[1] Allen, B. and M. Hellwig (1993). ”Bertrand-Edgeworth Duopoly with Proportional
Residual Demand,”International Economic Review, 34(1), 39-60.
[2] Anderson, S. P. and A. de Palma (1992). "The Logit as a Model of Product Di¤erentiation," Oxford Economic Papers, 44(1), 51-67.
[3] Beckmann, M. (1965). ”Edgeworth-Bertrand Duopoly Revisited,” in R. Henn, ed.,
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