Download Price choices, quantities and one mixed strategy equilibrium: A

Price choices, quantities and one mixed strategy
equilibrium: A review of classical duopoly models
by Daniel Cracau∗
Preliminary version, July 2012
Please do not distribute or cite.
Abstract
In this article, we present a systematic overview of the market outcomes for the classical oligopoly games. The study contains six games:
pure quantity competition, pure price competition as well as price and
quantity competition, each with simultaneous and sequential moves. We
present equilibrium prices, quantities and profits for each game. In particular, we present the mixed strategy equilibrium of the simultaneous
price and quantity competition. This equilibrium was found by Gertner
(1986), but remained unrecognized so far. From a comparison of market
outcomes, we can classify the situations where the choice of the decision
variable(s) and timing gives strategic advantages. Keywords:Mixed Strategy Equilibrium; Price; Quantity; Oligopoly Competition
1
Introduction
The search for the mixed strategy equilibrium in the oligopoly competition with
prices and quantities and simultaneous moves (PQ game) has puzzled many researchers, including myself, for a long time. To the best of our knowledge,
Shubik (1955) was the first to systematically investigate duopoly settings with
firms choosing prices and quantities at the same time. Since then, the literature
dealing with the PQ game can be divided into two strands. While one part of
the research underlines the non-existence of a pure strategy equilibrium, see for
example Friedman (1988), the other part of the research analyzes slightly modified versions of the game or elaborates specific properties of the equilibrium.
Shubik (1955) formulated the analytical framework for the analysis of the mixed
strategies in the PQ game. However, he was not able do derive the solution for
the equilibrium at this time. Later on, Levitan and Shubik (1978) introduce a
cost for unsold production, i.e. a constant loss for each unit of quantity exceeding actual demand. They are able to derive a precise definition of the mixed
strategy equilibrium then. The equilibrium is characterized by quantities being
a unique function of the price, a positive probability for each duopolist to stay
out of the market and zero expected profits. For the general PQ game (where
all production is costly), van den Berg and Bos (2011) generalize the finding of
∗ University of Magdeburg, Faculty of Economics and Management, Universitaetsplatz 2,
39106 Magdeburg, Germany, [email protected]
1
Tasnádi (2004) that firms make zero expected profits in equilibrium.
A precise definition of the mixed strategy equilibrium for the general PQ game
does, however, exist. Even more surprisingly, it has been found over 25 years
ago. The dissertation project entitled ”Essays in Theoretical Industrial Organization” by Robert H. Gertner (1986) includes an article on ”Simultaneous Move
Price-Quantity Games and Equilibrium without Market Clearing”. In this article, Gertner (1986) presents a general solution for the oligopoly competition
with firms choosing prices and quantities at the same time. He analyzes decreasing and constant marginal cost as well as increasing marginal cost. For the
first two cases, Gertner (1986) shows that in equilibrium the quantity in each
decision is a unique function of the price. Thus, the mixed strategy equilibrium
is completely represented by the probability distribution over prices. For the
case of increasing marginal cost, he applies a simulation, because no analytical
solution for a mixed strategy equilibrium in this game can be derived.
We will use his findings and compare them to the classical oligopoly models with
price and quantity as decision variables and simultanoues as well as sequential
moves of firms. The benchmarks are therefore Cournot (1838), Bertrand (1883),
von Stackelberg (1934) and Gelman and Salop (1983). To make the overview
tractable, we focus on the classical oligopoly models and thus have to exclude
variations of the classical settings. In particular, there are missing: (i) capacity constrained oligopolies as in Edgeworth (1897) or Levitan and Shubik
(1972), (ii) two-stage oligopolies with subsequent price and quantity choices as
in Kreps and Scheinkman (1983) or Friedman (1988) and (iii) mixed oligopolies
with some firms choosing prices and others choosing quantities as modeled in
Tasnádi (2006). For capacity contrained oligopolies, the equilibrium outcome
strongly depends on the degree of capacity limitation. Maskin (1986) generalizes the finding of Levitan and Shubik (1972), who show that possible equilibria
are either Bertrand or Cournot or mixed strategy equiilibria. For the twostage duopolies, Friedman (1988) proves that each equilibrium including pure
strategies is the same as the one in a game without the second stage, i.e. he
generalizes the findings of Kreps and Scheinkman (1983). Finally for mixed
oligopolies, Tasnádi (2006) derives that the equilibrium depends on the number
of price setting firms as well as on the firms’ capacity restrictions.
The overall goal of this article is to provide an overview about the classical models of oligopolistic interaction. Hereby, our two main contributions are: (i) we
formulate the explicit form of the mixed strategy equilibrium in the PQ game
in the classical setting of identical firms with constant marginal cost and (ii) we
evaluate the market outcomes in the PQ game in comparison to those in the
five classical duopoly games. With these two contributions, we would like to fill
the gap in the literature on classical oligopoly competition.
For this study, we present the mutually shared game theoretic framework of all
these models in Section 2. Then, we derive the equilibrium outcomes for six
different games, summarizing individual prices, quantities and profits. Hereby,
Section 3 includes the three games where decisions are all simultaneous whereas
the corresponding sequential games are subsequently presented in Section 4.
The equilibrium results are compared in Section 5 of this article. The final
2
section briefly discusses limitations and concludes.
2
Model preliminaries
We analyze two firms i = 1, 2 that compete in a single market. Firms’ decision
variables are either sales prices pi and/or production quantities qi . Demand is
a function of both firms’ prices and is thus generally given by D(pi , pj ) with
∂D(pi )
< 0. Total cost include production cost Ci (qi ) and fixed cost Fi . In
∂pi
general, a firm’s profit is therefore given in (1).
πi = pi min [qi , D(pi )] − Ci (qi ) − Fi
(1)
Table 1: Overview of simplified assumptions
General
assumption
Demand
Production cost
Fixed cost
Simplification
a
D(pi ) = b+d
−
Ci (qi ) = cqi
Fi = 0
D(pi )
Ci (qi )
Fi
b
b2 −d2 pi
+
d
b2 −d2 pj
For reasons of tractability, we will mainly consider the simplifying assumptions
displayed in Table 1 and apply them to the models. Firms are identical and
face linear production cost Ci (qi ) = cqi with no fixed cost (F1 = F2 = 0). The
demand is a linear function of prices and can be derived according to the idea
of a representative consumer maximizing his net payoff CR. The optimization
calculus is given by
max
qi ,qj
CR = U −
j
X
pk qk
(2)
qi2 + qj2 b + 2dqi qj
(3)
k=i
with U being the quadratic utility function
U = (qi + qj )a −
1
2
presented in Singh and Vives (1984) and i 6= j. The utility function was originally adapted from Dixit (1979). With given prices, the consumer’s net payoff maximization yields the well known linear inverse demand function pi =
a − bqi − dqj . The corresponding linear demand function can be derived as
a
b
d
1
D(pi ) = qi = b+d
− b2 −d
Hereby, parameter a indicates the high2 pi + b2 −d2 pj .
est willingness to pay in the market This value is also known as the prohibitive
1 Note that in the classical models there is no discrepancy between production and demand.
This is due to the fact that in quantity setting models (Cournot and Stackelberg), prices are
assigned by an ”omniscient auctioneer”. In price setting models, firms choose their own
quantities after demand is realized, but it is in their interest to meet demand.
3
price, i.e. the maximum price with positive demand. Parameter b measures
the relation between a firm’s price and its own demand, whereas d measures
the relation between a firm’s price and its competitor’s demand. In particular,
own-price elasticity and cross-price elasticity are given as
∂qi pi
∂pi qi
∂qi pj
j =
∂pj qi
bpi
,
(a − pi ) b − (a − pj ) d
dpi
=
.
(a − pi ) b − (a − pj ) d
=−
i =
For our comparison, we will consider the case of homogeneous goods (d → b)
only.2 In this case, the demand function simplifies to D(pi ) = a/b − pi for
pi < pj , i.e. the low price firm can serve the entire demand. The high price
firm (pi > pj ) can only serve residual demand, if there is any. In case of equal
prices, firms share the demand equally. For this article, we assume efficient
rationing which is relevant to the two games where firms decide on their prices
and quantities.3
As the results remain qualitatively unchanged, we assume w.l.o.g. that a = 1,
b = d = 1 and thus 0 ≤ c ≤ 1. These assumptions inprove the readability of the
comparison in the next sections.
3
3.1
Simultaneous Move Games
The Quantity Game
The game which is described here, is the one which is typically referred to as the
Cournot competition, see Cournot (1838). Both firms simultaneously choose
their production level qi . Total production Q = q1 + q2 then determines the
market clearing Cournot prices pC (Q). Firms’ profits can therefore be derived
!
i
as πi = qi pC − Ci (qi ) − Fi . Profit maximization ∂π
∂qi = 0 yields (4), which
incorporates firm i’s best response towards any of firm j’s production.
pC + qi
∂pC
dCi
−
=0
∂qi
dqi
(4)
Applying the simplifications, the symmetric Nash equilibrium is given through
1−c
,
3
1 + 2c
p1 = p2 = pC =
,
3
(1 − c)2
π1 = π2 = π C =
.
9
q1 = q2 = q C =
2 To the best of our knowledge, for the two games where simultaneous price and quantity
setting is assumed, no product differentiation has been modeled so far.
3 Davidson and Deneckere (1986) discuss different rationing rules. In general, the choice of
the rationing rule can have a main impact on the equilibrium of an oligopoly game. In our
games, however, it does not.
4
3.2
The Price Game
This game is usually referred to as the Bertrand competition, see Bertrand
(1883). In contrast to the previous quantity competition model, both firms decide simultaneously on their prices pi . The market clearing Bertrand quantities
qiB are then determined by the demand function D(pi , pj ). The corresponding
profit of firm i is then represented by πi = pi qiB − Ci (qiB ) − Fi . Considering the
first order condition for maximized profits
qiB + pi
∂πi
∂pi
!
= 0, (5) can be derived.
∂qiB
∂Ci
−
=0
∂pi
∂pi
(5)
Using our simplified assumptions, the symmetric Nash equilibrium can be derived as
p1 = p2 = pB = c ,
1−c
q1 = q2 = q B =
,
2
B
π1 = π2 = π = 0.
3.3
The Price-Quantity Game
The PQ game differs substantially from the two games analyzed before, as now
both price pi and quantity qi are chosen by the two firms. It was first analyzed
in Shubik (1955). In the PQ game, the firm i with the lower price sells its full
output qi up to the market demand D(pi ). The firm j (j 6= i) that decided on
the higher price can now satisfy the residual demand, which is given through
the efficient rationing rule
D(pj |pi ) = D(pj ) − si ,
where si is the amount sold by the lower-price competitor i.4 For the case of
equal prices (pi = pj ), the market demand is shared equally between the firms,
as far as the quantities qi allow, i.e. asymmetric market shares may arise with
equal prices if one firm’s quantity is too low to meet its demand. These rules
can be summed up as in Gertner (1986) by the following equation for the sales
si of firm i,

min[qi , D(pi )]
, if pi < pj ,





min [qi , D(pi ) − sj ]
, if pi > pj ,
si (p1 , q1 , p2 , q2 ) =
(6)

h
n
oi



 min qi , D(pi ) − min qj , D(pj )
, if pi = pj .
2
4 For the model presented here, Gertner (1986) establishes that the main properties of
the equilibrium are not affected when efficient rationing is assumed instead of proportional
rationing.
5
It follows that in contrast to classical oligopoly games, it is possible in this
game to have qi 6= Di (pi ), i.e. firms may choose a quantity that is too high
or too low to fit their demand. Production cost, however, arise according to qi
even in case of qi > D(pi ), i.e. production is always costly independent from
actual sales. Correspondingly, the profit of firm i is generally given through
πi = pi si − Ci (qi ) − Fi .
Gertner (1986) explains that a pure-strategy equilibrium does not exist in this
game.5 However, he proves that there exists (at least) one mixed strategy
equilibrium, i.e. each of the firms strategy can be described by the probability
density function fi (pi , qi ) that states the probability of firm i to play the strategy
(pi , qi ). According to Shubik (1959) the probability density function f1 (p1 , q1 )
and f2 (p2 , q2 ) form a mixed strategy equilibrium, if the integrals
Z ∞Z ∞
V̄i =
πi (p1 , q1 , p2 , q2 )dfj (pj , qj ) ,
0
0
are constant for all strategies (pi , qi ) with positive probability according to
fi (pi , qi ). Shubik (1959) refers to V̄i as the value of the game for firm i, i.e.
the maximum profit it can achieve if the strategy of the opposition firm is
known. Note that in the case of the symmetric game considered here, Gertner
(1986) has shown that the mixed strategy equilibrium is also symmetric, which
means f1 ≡ f2 .
For a game of this type with the assumption of non-increasing marginal cost,
Gertner (1986) proves that all Nash equilibria satisfy V̄i = 0 and that every
price quantity choice with a positive probability in this equilibrium satisfies
qi = D(pi ). Therefore, the probability distribution over all price choices completely describes the equilibrium.
The symmetric mixed-strategy equilibrium derived in Gertner (1986) is given
through

, for pi ∈ [0, c) ,
 0
1 − c/pi , for pi ∈ [c, p̂) ,
(7)
F (pi ) =

1
, for pi ∈ [p̂, ∞)
where p̂ is the prohibitive price, i.e. D(pi ≥ p̂) = 0. Moreover, he has proven
this equilibrium is unique, if marginal cost are constant. These equilibrium
characteristics are well aligned with the findings of Levitan and Shubik (1978).
We interpret this equilibrium in the following way: both firms choose their prices
independently of each other from a range between the marginal cost level and
the prohibitive price. Firms have two basic choices: (i) with probability 1 − c/p̂
each, a firm enters the competition. Among prices below the prohibitive one,
lower prices are chosen with a higher probability; (ii) with probability c/p̂ each,
a firm charges the prohibitive price and produces nothing. Thus, this firm does
not enter the competition in this case. For the case with two firms participating
5 In particular, the Bertrand equilibrium with prices at the marginal cost level is not an
equilibrium. In this case the other firm would be better off earning positive profits by acting
as a residual monopolist rather than committing to the Bertrand equilibrium.
6
(pi < p̂ for i = 1, 2), the lower price firm earns a positive profit, whereas the
other firm faces a loss equal to its production cost.6 In expected terms, profits
are equal to zero in this static game.
Applying the simplifications of linear demand and cost functions, the mixed
strategy equilibrium can be described by its distribution function

, for pi ∈ [0, c) ,
 0
1 − c/pi , for pi ∈ [c, 1) ,
F (pi ) =
(8)

1
, for pi ∈ [1, ∞)
and the quantity function qi = 1 − pi . To compare the mixed strategy equilibrium with the pure strategy equilibria in the other games, we calculate expected
values and variances of the equilibrium outcomes.
Proposition 1. The mixed strategy equilibrium in the PQ game with the simplifying assumptions of demand following D(p) = 1 − p and constant marginal
cost c with no fixed cost satisfies
c
ln(c) ,
c−1
c2
Q
Var[pP
ln(c)2 ,
i |pi < 1] = c −
(1 − c)2
c
E[qiP Q |qi > 0] = 1 −
ln(c) ,
c−1
c2
ln(c)2 ,
Var[qiP Q |qi > 0] = c −
(1 − c)2
Q
E[pP
i |pi < 1] =
E[πiP Q ] = 0 ,
Var[πiP Q ] =
c2
−5 + 4c + c2 − 2(1 + 2c) ln(c) .
2
Proof. See Appendix A.
4
4.1
Sequential Move Games
The von Stackelberg Game
This game includes two firms deciding on their quantities qi as in the Cournot
game, but in an exogenous order. It was first modeled in von Stackelberg
(1934). We assume that firm i = 1 (referred to as being the leader ) decides
before firm i = 2 (referred to as being the follower ). After both firms have
chosen their quantities, the market clearing von Stackelberg prices pS (Q) are
determined by total production. Thus, firm i’s profit can be calculated as
6 In the continuous PQ game described here, the possibility that both firms charge the same
price (except for the prohibitive price) is zero.
7
πi = qi pS − Ci (qi ) − Fi . The follower maximizes his profit by considering
the leader’s quantity choice q¯1 . Therefore, the follower’s best response quantity
choice q2BR (q¯1 ) can be derived from the profit maximizing condition given by (9).
pS (q¯1 ) + q2
∂pS (q¯1 ) ∂C2
−
=0
∂q2
∂q2
(9)
The leader will anticipate the follower’s best response and therefore chooses a
!
1
quantity subject to ∂π
∂q1 = 0. The condition for his profit maximizing quantity
is then given in (10).
pS + q1
∂C1
∂pS
−
=0
∂q1
∂q1
(10)
Finally, the symmetric Nash equilibrium including the simplified assumptions
is given through
q1 = q1S =
q2 = q2S =
p1 = p2 = pS =
π1 = π1S =
π2 = π2S =
4.2
1−c
,
2
1−c
,
4
1 + 3c
,
4
(1 − c)2
,
8
(1 − c)2
.
16
The Price Leadership Game
The game described here is comparable to the previous one and was also analyzed first in von Stackelberg (1934). Two firms decide sequentially, but now
the strategic decision variables are their prices pi . Again, we will consider the
first moving firm i = 1 as the leader and the second moving firm i = 2 as the
follower. The corresponding market clearing Price Leadership quantities qiP L
are determined by the demand function D(pi ) afterward. A firm i’s profit is
therefore given as πi = pi qiP L − Ci (qiP L ) − Fi . Taking the leader’s price choice
p¯1 into account, the follower maximizes his profit by choosing a price subject to
∂π2 !
BR
∂p2 = 0. The first order condition for the follower’s best response price p2 (p¯1 )
is displayed in (11).
q2P L (p¯1 ) + p2
∂q2P L (p¯1 ) ∂C2
−
=0
∂p2
∂p2
(11)
As the leader can forecast the follower’s reaction, he includes this response into
!
1
his optimization calculus and chooses a price subject to ∂π
∂p1 = 0. The corresponding condition for the leader’s optimal quantity can therefore be derived
8
as (12).
q1P L + p1
∂q1P L
∂C1
−
=0
∂p1
∂p1
(12)
Applying our simplifications, the symmetric Nash equilibrium in this game can
be derived as
L
p1 = p2 = pP
= c,
1
3
q1 = q1P L = (1 − c) ,
8
5
q2 = q2P L = (1 − c) ,
8
π1 = π2 = π1P L = 0
4.3
The Judo Game
The game with simultaneous price and quantity setting and firms deciding one
after another was analyzed in detail in Gelman and Salop (1983). The basic
structure of the equilibrium was, however, first mentioned in Levitan and Shubik
(1978). The equilibrium is mainly characterized by the leader’s decision, which
was described by Gelman and Salop (1983) in the following manner: ”To capture
the image of a small firm using its rival’s large size to its own advantage, we
call this a strategy of judo economics” (Gelman and Salop, 1983, p. 315). We
therefore refer to the game as the Judo game.
We denote the leader in this game as i = 1 and the follower as i = 2. First, the
leader chooses a price p1 and a quantity q1 . The follower then decides on his
price p2 as well as on his quantity q2 . Optimally, the follower will always choose a
quantity that fits with his demand. It is therefore sufficient to consider his price
choice only and to assume that his quantity decision follows the corresponding
demand. The setting of the Judo game leads to two possible scenarios after
the follower’s price choice. If the follower marginally undercuts the leader, i.e.
p2 = lim→0 p1 − = p1 , he becomes the sole firm serving the market at a price
p1 . In this case, the follower’s profit equals
π2und = p1 D(p1 ) − C2 (q2 ) − F2
with q2 = D(p1 ). The leader faces a loss equal to the sum of his production
cost and his fixed cost if he is undercut. Otherwise, if the follower chooses
not to undercut the leader, he serves residual demand. As in Gelman and
Salop (1983), we apply efficient rationing to calculate residual demand, i.e. the
consumers with the highest willingness to pay are served first.7 Thus, residual
demand is given by Dres = D(p2 ) − q1 in this case. The follower’s profit can be
calculated as
π2res = p2 (D(p2 ) − q1 ) − C2 (q2 ) − F2
7 Gelman and Salop (1983) show that the equilibrium solution also holds for proportional
rationing rules.
9
with q2 = D(p2 ) − q1 . The leader’s profit equals π1 = p1 q1 − C1 (q1 ) − F1 .
The leader now faces the problem to decide on a price-quantity pair that maximizes his profit, i.e. a combination of price p1 and quantity q1 so that he will
not be undercut by the follower. Therefore, he should guarantee for
π2und ≤ π2res
(13)
in equilibrium. We introduce θ(q1 ) as the function that gives the highest possible
price p1 according to the leader’s quantity choice q1 so that (13) holds with
equality. The leader’s optimization problem is finally given as
max
p1 ,q1
π1 = p1 q1 − C1 (q1 ) − F1
s.t.
(14)
p1 ≤ θ(q1 )
The corresponding condition for the leader’s optimal quantity is thus given
in (15).
θ(q1 ) + q1
dθ
dC1
−
=0
dq1
q1
(15)
The leader’s optimal price follows p1 = θ(q1 ). The follower’s reaction is derived
as p2 = argmax π2res (q1 ) and q2 = Dres (p2 ).
When we apply the simplification as presented in the assumptions section, we
can derive the following Nash equilibrium outcomes for the Judo game:
√
(2 + 2)(1 + c)
J
,
p1 = p1 =
√ 4
2− 2
q1 = q1J =
(1 − c) ,
2
1−c
p2 = pJ2 = √ + c ,
8
(1 − c)
J
q2 = q2 = √
,
8b
√
(2 − 2)2
J
π1 = π1 =
(1 − c)2 ,
8
(1 − c)2
π2 = π2J =
.
8
5
Comparison of Profits and Market Efficiency
The models described in this article illustrate the basic ways how competition
with prices or quantities can be modeled in duopoly settings. We presented the
symmetric Nash equilibria for six games. In this section, we will compare the
games with respect to prices, quantities and profits.
We define total quantity Q as the sum of the two firms’ quantities (Q = q1 + q2 )
and total profits Π as the sum of the two firms’ profits (Π = π1 +π2 ). Table 3 (at
10
Equilibrium price of the PQ game
E[pi|pi<1]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cost parameter (c)
Figure 1: Equilibrium price of the mixed strategy equilibrium in the PQ game
(solid line) vs monopoly price (dashed line)
the end of the article) displays individual as well as collective quantities, prices
and profits in the six previously analyzed games. In the sequential games, index
1 denotes the leader whereas index 2 denotes the follower. As a benchmark,
also the shared monopoly outcome is presented, i.e. the market situation with
cooperating firms.
Looking at the equilibrium prices, one can see that the Bertrand price and
the Price Leadership price are equal to marginal cost with our simplified game
parameters. The next highest price is that of the Judo leader, followed by the
von Stackelberg price, the Cournot price and finally the Judo follower’s price.
All prices are below the monopoly price pM .
pB = pP L = c < pJ1 < pS < pC < pJ2 < pM
The ranking of the expected price in the PQ game depends on the cost parameter. Figure 1 depicts the expected price of the mixed strategy equilibrium as
a function of c. In Table 2, we have also indicated the critical values c∗ from
which on E[pP Q ] exceeds the equilibrium prices of the other games, i.e. c∗ for
c ≤ c∗ ↔ equilibrium price ≥ E[pP Q ].8 It turns out, that for very low values
of c, E[pP Q ] is the lowest of all equilibrium prices, while for high values of c, it
is the highest, converging towards the shared monopoly price.
Comparing the equilibrium quantities, a similar pattern holds. Except for the
von Stackelberg follower, all equilibrium quantities exceed the shared monopoly
8 Under our simple game parameter, the equilibrium prices of all other five games are linear
increasing in c. Thus, if E[pP Q ] is higher than one of these equilibrium prices at c∗ , it is also
higher for all other c > c∗ .
11
Table 2: Critical values for E[pP Q ] in comparison to equilibrium prices
equilibrium price
critical cost parameter c∗
pJ1
pS
pC
pJ2
≈ 0.04251664921
≈ 0.1954781652
≈ 0.3549846535
≈ 0.4055504829
Equilibrium quantity of the PQ game
E[qi|qi>0]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cost parameter (c)
Figure 2: Equilibrium quantity of the mixed strategy equilibrium in the PQ
game (solid line) vs monopoly quantity (dashed line)
quantity q M/2 and can be ranked as follows:
q M/2 = q2S < q1J < q C < q2J < q1P L < q1S = q B < q2P L .
Again, figure 2 depicts the expected quantity of the mixed strategy equilibrium as a function of the cost parameter c. For all cost parameter c < c∗ ≈
0.4655227174, the quantity in the PQ game is the highest of all games. For
c > c∗ the PQ quantity ranks second highest after q2P L .
Individual profits are found to be zero in the Bertrand game as well as in the
Price Leadership game. This finding can be directly inferred from the equilibrium prices in these three games which are equal to marginal cost. In the PQ
game, expected profits are also zero. In contrast, the von Stackelberg leader as
well as the Judo follower earn shared monopoly profits. The other individual
profits are in between.
π B = π P L = π P Q = 0 < π1J < π2S < π C < π S = π2J = π M/2
Total quantity exceeds the monopoly quantity QM in all games. In the Bertrand
as well as in the Price Leadership game, total quantity equals the full market
12
size (which can be served at prices equal to marginal cost). In the PQ game
total quantity even is greater than the market size.
QM < QJ < QC < QS < QJ < QB = QP L < QP Q .
Looking at the total profits in the different games, one sees that all of them are
below the monopoly profit ΠM . The highest total profit can be found in the
Cournot game, followed by the von Stackelberg game and the Judo game. Zero
total profits occur in the Bertrand game, the PQ game and the Price Leadership
game. The ranking therefore follows
ΠB = ΠP Q = ΠP L = 0 < ΠJ < ΠS < ΠC < ΠM .
These rankings provide a good basis to summarize the main properties of the
games. In the Bertrand game and the Price Leadership game the price competition among the two firms is so strong that prices converge to the marginal
cost level, when products are homogeneous. This finding is best known for the
Bertrand game, but it is also obvious that a firm in the follower in the Price
Leadership game has a strong incentive to always undercut its rival. For the
PQ game, expected profits are zero indepeedntly of the game parameters. In
the Judo game, however, the leader uses the quantity limitation to minimize
the follower’s incentive to undercut him by leaving a large residual demand. We
therefore conclude that in games with an established price competition both
firms would favor a sequential price setting with an opportunity of production
commitment.
In contrast, if only the timing of the game is exogeneous, a leader would prefer
a pure quantity choice whereas a follower would prefer a price-quantity choice.
If the game is known to be simultaneous, both firms favor the pure quantity
competition. Finally, if a quantity competition is fixed part of the game, each
firm favors being the Stackelberg leader over being in the Cournot competition. In a sequential quantity game, however, the follower prefers the leader to
additionally choose a price.
6
Conclusion
The explicit form of the mixed strategy equilibrium in the PQ game has not
been in the scientific discourse so far. Therefore, this comparison is the first
comprehensive overview of its kind. It shall offer a workhorse model for all
researchers working on variations of these games. In particular, we derived the
probability distribution over prices which completely define the mixed strategy
equilibrium. We also calculated expected values and variances of prices, quantities and profits for the PQ game. Using these calculations, we were able to give
a complete comparison of the classical duopoly games with linear demand and
constant marginal cost. With this work, we hope to fill the gap in the literature
on classical oligopoly competition.
A possible reason for the fact that an explicit solution for the mixed strategy
13
equilibrium in the PQ game has been derived relatively late is the model approach of previous works. In Shubik (1955), Shubik (1959) and Levitan and
Shubik (1978) the game was analyzed under the assumption of zero production
cost. Usually, the zero cost assumption is a useful simplification to focus on specific equilibrium properties. In this game, however, there is no mixed strategy
equilibrium under the zero cost assumption, as both firms then commit to zero
prices and each produces the entire market demand.
Compared to the five classical oligopoly games with pure strategy equilibria,
the mixed strategy equilibrium in the PQ game incorporates two characteristic
properties. First, it includes non market clearing quantities. As both firms
always choose to produce the entire demand at their chosen price, there is overproduction, given both firms are in the market. Second, expected profits are
zero although average prices are above cost. This is due to the losses of the firm
with the higher price.
The main results of this article were presented for duopolies only. However, the
insights can be expanded to n-firm oligopolies straight forward. In particular,
Gertner (1986) shows that the results of the mixed strategy equilibrium in the
PQ game are similar for any discrete number of firms. A further assumption
that was made during the analysis was that of identical firms. The introduction
of cost asymmetry may dramatically change the equilibria in the games with
price competition. This will happen if a cost advantage results in a corner solution with one firm pricing the other firm out of the market in equilibrium.
A natural extension of the research on oligopolies is the modeling of endogeneous
entry as introduced by Etro (2006). For the PQ game, however, endogeneous
entry is hardly plausible as the expected profits are already zero for the minimum number of two competitors. Studying a two-stage game, where entry takes
place before an oligopoly competition seems more appropriate. The analysis of
the PQ game in a dynamic setting provides a further direction for subsequent
theoretical work.
Overall, this article aims on helping to complete the theoretical oligopoly research rather than providing explicit policy implications. Answering the question how to come up with optimal price and quantity decisions in reality lies
beyond this analysis and will provide an additional starting point for further
(more empirically or experimentally oriented) research.
Acknowledgements
I greatly appreciate the constructive comments from Iwan Bos and Benjamin
Franz.
References
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Journal des Savants 67, 499–508.
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des Richesses. Paris: Hachette (translated by N. Bacon, New York: Macmillan
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Gelman, J. R., Salop, S. C., 1983. Judo economics: Capacity limitation and
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URL http://dspace.mit.edu/handle/1721.1/14892
Kreps, D. M., Scheinkman, J. A., 1983. Quantity precommitment and bertrand
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American Economic Review 76 (2), 382–386.
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Econometrica 23 (4), 417–431.
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Sons, Inc.
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duopoly. RAND Journal of Economics 15 (4), 546–554.
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of Economic Behavior & Organization 54 (2), 191–204.
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Tasnádi, A., 2006. Price vs. quantity in oligopoly games. International Journal
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von Stackelberg, H., 1934. Marktform und Gleichgewicht. Berlin: Springer.
A
Proof of Proposition 1
Proposition 1 The mixed strategy equilibrium in the PQ game with the simplifying assumptions of demand following D(p) = 1 − p and constant marginal
cost c satisfies
c
ln(c) ,
c−1
c2
Q
Var[pP
|p
<
1]
=
c
−
ln(c)2 ,
i
i
(1 − c)2
c
E[qiP Q |qi > 0] = 1 −
ln(c) ,
c−1
c2
ln(c)2 ,
Var[qiP Q |qi > 0] = c −
(1 − c)2
Q
E[pP
i |pi < 1] =
E[πiP Q ] = 0 ,
Var[πiP Q ] =
c2
−5 + 4c + c2 − 2(1 + 2c) ln(c) .
2
Proof. For this appendix we denote pi as p, qi as q and πi as π, if appropriate.
As the equilibrium is symmetric, this notation makes the proof more readable.
According to (7), the distribution function of the prices for a = 1 and b = 1 is

, for p ∈ [0, c) ,
 0
1 − c , for p ∈ [c, 1) ,
F (p) =

1
, for p ∈ [1, ∞).
Hence, the corresponding density function takes the form
c/p2 , for p ∈ [c, 1) ,
f (p) =
0
, otherwise .
16
(16)
The PQ prices
To derive E[p|p < 1], i.e. the conditional expected price for the case where
firm i decides to engage in the duopoly competition, we have to rescale f (p) to
R1
guarantee that 0 f (p|p < 1) dp = 1. Therefore, we get
f (p|p < 1) = f (p)
1
.
1−c
(17)
The conditional expected price can now be derived as
Z 1
c
pf (p|p < 1) dp =
ln(c) .
E[p|p < 1] =
c
−
1
0
Correspondingly, for Var[p|p < 1] we obtain
Z
Var[p|p < 1] =
1
p2 f (p|p < 1) dp − E(pi |pi < 1)2 = c −
0
c2
ln(c)2 .
(1 − c)2
The PQ quantities
To derive E[q|q > 0], i.e. the conditional expected quantity for the case where
firm i decides to engage in the duopoly competition, we follow the same reasoning. It is known from Gertner (1986) that in equilibrium q(p) = 1 − p, i.e.
quantity is a unique function of the price and always equals demand. Hence,
Z 1
c
E[q|q > 0] =
q(p)f (p|p < 1) dp = 1 − E[p|p < 1] = 1 −
ln(c) .
c
−
1
0
Correspondingly, Var[q|q > 0] can be derived as
Z
1
Var[q|q > 0] =
q(p)2 f (p|p < 1) dp − E[q|q > 0]2 = Var[p|p < 1] ,
0
=c−
c2
ln(c)2 .
(1 − c)2
The PQ profits
According to Gertner (1986), the profits satisfy E[π] = 0. For the derivation of
Var[π] we need to consider the three situations that may arise depending on the
firms’ decision to stay out of the market:
(i) Firm i is in the market and firm j is out of the market. In this case, firm
i earns πi = (pi − c)(1 − pi ), i.e. a profit according to the price and quantity
chosen. It follows
Z 1
E(i) [πi ] =
(pi − c)(1 − pi )f (p|p < 1) dp .
0
17
(ii) Firm i is out of the market and makes zero profit, i.e.
E(ii) [πi ] = 0 .
(iii) Both firms i and j are in the market. If pi < pj , firm i earns πi = (pi −
c)(1 − pi ), i.e. a profit according to the price and quantity chosen. If pi > pj ,
firm i earns πi = −c(1 − pi ), i.e. a loss according to the production cost. It
follows
Z 1Z 1
(iii)
(pi − c)(1 − pi )f (pj |pj < 1) dpj f (pi |pi < 1) dpi
E
[πi ] =
Z
0
1
Z
pi
pi
c(1 − pi )f (pj |pj < 1) dpj f (pi |pi < 1) dpi .
−
0
0
The expected value over all three situations can be derived by weighting the
situations according to their probabilities:
E[πi ] = (1 − c) c E(i) [πi ] + (1 − c) E(iii) [πi ] + c E(ii) [πi ] = 0 .
The same reasoning holds for the variance, where we obtain
Var[πi ] = (1 − c) c Var(i) [πi ] + (1 − c) Var(iii) [πi ] + c Var(ii) [πi ] ,
=
c2
−5 + 4c + c2 − 2(1 + 2c) ln(c) ,
2
where
Var(i) [πi ] =
Z
1
2
((pi − c)(1 − pi )) f (pi |pi < 1) dpi − E(i) [πi ]2 ,
0
Var(ii) [πi ] = 0 ,
Z 1Z
(iii)
Var
[πi ] =
0
Z 1
1
2
((pi − c)(1 − pi )) f (pj |pj < 1) dpj f (pi |pi < 1) dpi
pi
Z pi
2
(−c(1 − pi )) f (pj |pj < 1) dpj f (pi |pi < 1) dpi
+
0
0
− E(iii) [πi ]2 .
18
Table 3: Summary of equilibrium outcomes in the six classical duopolies with price or quantity as decision variable (a = 1,
b = 1, d = 1)
pi
Shared
monopoly
Cournot
1+c
2
1+2c
3
19
qi
1−c
4
1−c
3
πi
(1−c)2
(1−c)2
8
9
Q
1−c
2
Π
(1−c)2
4
Bertrand
PQ
von
Stackelberg
c
c ln(c)
c−1
1+3c
4
1−c
2
1−
c ln(c)
c−1
1−c
2
1−c
4
(1−c)2
8
(1−c)2
16
q1 =
q2 =
π1 =
π2 =
Price
Leadership
Judo
c
p1 = (2− 2)(1+c)
4
√ +c
p2 = 1−c
8
q1 =
q2 =
3(1−c)
8
5(1−c)
8
0
0
0
2(1−c)
3
1−c
ln(c)
2 − 2 cc−1
3(1−c)
4
1
2(1−c)2
9
0
0
3(1−c)2
16
0
√
√
q1 = (2− 2)(1−c)
2
√
q2 = 1−c
8
√
2
π1 = (2− 2)8 (1−c)
2
π2 = (1−c)
8
√
(4− 2)(1−c)
4
√
(7−4 2)(1−c)2
8
2