Cournot and Bertrand-Edgeworth competition when
rivals’costs are unknown
Jason J. Lepore
June 26, 2008
Abstract
We study a two-stage game with capacity precommitment followed by price competition where …rms have incomplete information about their rival’s marginal cost.
The game has a Cournot outcome if and only if the lowest possible marginal cost is
su¢ ciently high relative to the expected marginal cost.
Keywords: Bertrand-Edgeworth, Cournot, Consumer rationing, Incomplete information.
JEL classi…cations: D21, D43, D82, L11, L13.
1
Introduction
Kreps and Scheinkman (1983) (hereafter K&S) prove that in a two-stage game where …rms
choose capacities followed by price, the unique Nash equilibrium has the Cournot outcome.
A natural question is: how does uncertainty about the rival …rm’s cost e¤ect the coincidence
with the Cournot model? The premise of this paper is to prove the necessary and su¢ cient
conditions for a two-stage game to have a Cournot outcome when …rms are uncertain about
their rival’s marginal cost of capacity. In addition, we characterize exactly when there is
A¢ liation: Department of Economics, Orfalea College of Business, California Polytechnic State University, San Luis Obispo, CA 93407; e-mail: [email protected]; phone: (805) 756-1618; fax : (805) 756-1473.
1
2
an equilibrium with a Cournot outcome for both e¢ cient and proportional rationed residual
demand.1
We show that the Cournot outcome is an equilibrium of the two-stage game if and only if
the Cournot equilibrium is such that at the lowest capacity cost realizations, …rms’capacities
lead to market clearing prices.2 The wider the dispersion of cost uncertainty, the greater the
lower bound cost must be for the K&S result to be possible.
To provide a concrete illustration of our results, we calculate an example with linear
demand and a uniform distribution over marginal costs. With e¢ cient rationing, there is an
equilibrium with a Cournot outcome if and only if three times the lower bound marginal cost
is weakly greater than the upper bound marginal cost, while under proportional rationing,
the bounds must be closer.
2
The Model Basics
Consider an industry with two …rms producing a single homogeneous product. The two …rms
compete in a game where they …rst choose capacities independently and simultaneously;
these choices are made public, then prices are chosen independently and simultaneously.
The market demand is D : R+ 7! R+ . The inverse demand is P : R+ 7! R+ . Demand is
bounded when the market price is zero, so X = D(0) < 1. The following assumptions are
maintained throughout the paper.
Assumption 1 (A1) P (x) is twice-continuously di¤erentiable, strictly positive, strictly decreasing, and concave 8x 2 (0; X). In addition, P (x) = 0, 8x X.
Assumption 2 (A2) 9 2 R+ r f1g such that
limx"X P 0 (x)
.
Each …rm’s capacity provides an absolute limit on the number of units it can produce:
its cost of production is zero up to capacity, and in…nite for any quantity beyond. Each …rm
faces a constant marginal cost of capacity.
Assumption 3 (A3) Each …rm’s marginal cost of capacity is independently drawn with
_
probability measure from A = a; a . has no mass points and is such that (A) = 1 and
(S) < 1, 8S A.3
1
K&S only address e¢ cient rationing.
Here Cournot outcome is used to refer to the Cournot outcome of a game with incomplete cost information.
3
The arguments of Lemma 1 and Theorem 2 are greatly simpli…ed based on the restrictions on . If we
allow to be a measure without full support, the primary content is una¤ected.
2
3
Assumption 4 (A4) Each …rm’s cost of capacity permits positive pro…t: a < P (0).
Next we move to characterizing the equilibria of the Cournot game with incomplete cost
information.
2.1
The Cournot Game
We …rst present the basic features of the Cournot game with uncertain costs (hereafter UC).
We take the generic …rm to have marginal cost a and capacity x. Denote by Q = (q ) 2A ,
the capacity choice for each possible rival with cost 2 A.
Denote …rm a’s UC pro…t function by
c
a (x; Q)
= E [P (x + q )x]
(1)
ax.
This function is de…ned for all (x; Q); it is twice continuously di¤erentiable and strictly
concave for all x 2 (0; X max 2A q ). Firm a’s UC best response to Q is
a (Q)
= arg max
x 0
c
a (x; Q).
(2)
Our …rst result is a condition under which the UC game has a unique equilibrium.
Theorem 1 In the UC game de…ned by A1-4, if there is an equilibrium Q = (qa )a2A such
_
_
that qa < X=2, then Q is the unique equilibrium such that qa < X=2.
_
For the case of linear demand, there is always a Q such that qa < X=2.
Proof. Notice that at such Q ; ca is di¤erentiable. Hence, based on the Kuhn-Tucker
Theorem the …rst order condition is necessary. Based on A1&A2 for all a 2 A the necessary
condition is shown below,4
E [P 0 (qa + q )qa + P (qa + q )]
a = 0.
(3)
First we show that if a0 > a, then qa0 < qa . Suppose to the contrary that qa0 > qa for a0 > a.
Based on the strict concavity of each revenue function P (x + q )x, 8 2 A, P 0 (qa0 + q )qa0 +
P (qa0 + q ) < P 0 (qa0 + q )qa0 + P (qa0 + q ). Hence, if condition (3) for cost a0 holds for qa0 ,
then it cannot also hold true for a at qa , a contradiction.
4
Di¤erentiation can be passed inside the expectation based on Billingsly (1986), Theorem 16.8.
4
_
To prove that there is only one equilibrium such that qa < X=2, we show that the best
_
Q
response a (Q) is a contraction mapping on a2A [0; X=2].5 If
2
@xx
then
a (Q)
c
a (x; Q)
P
2A
c
a (x; Q)
2
@xq
< 0,
(4)
is a contraction mapping. The expression in (4) can be rewritten as
E [P 00 (x + q )x + 2P 0 (x + q ) + jP 00 (x + q )x + P 0 (x + q )j] < 0.
(5)
Since, both terms in the absolute value are non-positive, (5) can be simpli…ed to E [P 0 (x + q )] <
0, which must be true based on A1.
2.2
The Pricing Subgames
Now let us begin to analyze the two-stage game. The character of the Nash equilibrium in
any pricing subgame will depend on …rms’capacities (x; y).
Before we move to the characterization of the Nash equilibria of pricing subgames, we
formally address the way in which demand is rationed.
Assumption 5 (A5) The residual demand is rationed according to either the e¢ cient or
proportional rule.
More speci…cally, the demand served by …rm i is:
8
>
< minfx;
n D(pi )g
n
oo if pi < pj
i)
Dir (p1 ; p2 ) =
min x; max D(p
; D(pi ) y
if pi = pj ;
2
>
:
r
min fx; max f0; di (p1 ; p2 ; y)gg
if pi > pj
(6)
where r 2 fe; pg, “e” represents “e¢ cient rationing” and “p” represents “proportional rationing.”The two residual demands for r = e and r = p are:
dei (pi ; pj ; y) = D(pi )
y,
and
dpi (pi ; pj ; y) = D(pi ) 1
y
D(pj )
.
Dasgupta and Maskin (1986) show that mixed strategy Nash equilibrium exist in all
pricing games. The characterization of the pricing subgames below follows from Lepore
5
See Vives (1999) Section 2.5 for a discussion of uniqueness of equilibrium and best response contraction
mappings.
5
(2008) which is based on results from K&S, Davidson and Deneckere (1986) and Deneckere
and Kovenock (1992). Label the Nash equilibrium mixed strategies ( ra ; rb ). In order to
delineate pricing regions, we de…ne the certain Cournot best response function for a …rm
with zero cost
_
r(y) = arg maxx2[0;X]
P (x + y)x.
Denoted by q c , the zero cost Cournot equilibrium capacity. Denote by q m , the zero cost
monopoly capacity.
For either rationing rule, equilibrium pricing is divided into three regions based on the
…xed capacities (x; y).
Bertrand pricing:
n
_
B = (x; y) 2 [0; X]
_
_
o
X .
[0; X] j minfx; yg
Cournot pricing:
C
p
Ce
n
_
= (x; y) 2 [0; X]
n
_
= (x; y) 2 [0; X]
Mixed strategy pricing:
n
_
Mr = (x; y) 2 [0; X]
_
[0; X] j x + y
_
[0; X] j x
r
o
,
r(y) and y
o
r(x) .
_
_o
[0; X] j (x; y) 2
= Cr , min fx; yg < X :
The Nash equilibrium unique expected revenue of
8
P (x + y)x
if
>
>
< r r
(p (y; x); y) if
Rr (x; y) =
r
p (x; y)x
if
>
>
:
0
if
where,
q
m
R
(p; y) = maxpa 2R+ pa dra (pa ; p; y)d rb .
…rm i is:
(x; y) 2 Cr
(x; y) 2 Mr and x y
(x; y) 2 Mr and x < y
(x; y) 2 B,
If (x; y) 2 Mr , then the expected revenue of each …rm is determined by the lowest price
in the support of mixed strategy of the smaller …rms. Denote by pr (x; y), this price when
x y.
2.3
The Capacity Choice Game
The capacity choice expected pro…t given Nash equilibrium pricing is de…ned as
r
a (x; Q)
= E[Rr (x; q )]
ax,
(7)
6
and the corresponding capacity choice best response of …rm a to Q is given by
r
a (Q)
= arg max
x2[0;X]
r
a (x; Q).
(8)
Note that the function ea (x; Q) is bounded and continuous. Hence, for any Q 2
it attains a maximum over all x 2 [0; X].
3
Q
a2A [0; X],
Results
We move to the primary result of the paper; a characterization for when the capacity choice
game has an UC outcome.
_
Theorem 2 Suppose that Q such that qa < X=2. In the capacity choice game de…ned by
A1-5, the UC capacities are an equilibrium if and only if
qa ; qa 2 Cr .
(y)
As a preliminary result, we establish that the right-hand derivative of the expected pro…t
being equal to zero is necessary for Q to be an equilibrium of the capacity choice game.
For notational convenience, we write the …rst partial derivative of a function f (x; y) in
the argument x at (x; y) as @x f (x; y) and right-hand and left-hand partial as @x+ f (x; y) and
@x f (x; y), respectfully.
Lemma 1 Q can be an equilibrium of the capacity choice game only if
@x+
r
a (qa ; Q)
= 0, 8a 2 A.
Proof. First, since ra (qa ; Q) is Lipschitz continuous, it is di¤erentiable almost everywhere
and thus for all (qa ; Q) the right-hand derivative always exists.
Suppose to the contrary that 9a 2 A such that @x+
r
a (qa ; Q)
6= 0 and Q is an equilibrium.
If @x+ ra (qa ; Q) > 0, then an arbitrarily small increase in capacity is preferred to qa , a
contradiction.
If @x+
r
a (qa ; Q)
< 0, then based on A1&2, we can rewrite @x+
E @x+ Rr (qa ; q )
r
a (qa ; Q)
as
a < 0:
Only at symmetric capacities x = y can Rr (x; y) be non-di¤erentiable, hence only at (qa ; qa ),
might @x+ Rr (qa ; qa ) 6= @x Rr (qa ; qa ). Since, the probability measure has no mass points and
7
@x+ Rr (qa ; qa ) and @x Rr (qa ; qa ) are bounded, E [@x+ Rr (qa ; q )] = E [@x Rr (qa ; q )]. Thus, the
left-hand derivative at Q is negative, which implies an arbitrarily small increase in capacity
is preferred to qa , a contradiction.
Before proceeding to the proof of Theorem 2 we restate some facts established in Lepore
(2008) which will be necessary for our proof. First, de…ne
pep (x; y) = minfp
R
0 j p(y ^ D(p)) = pm drb (pm ; p; x)d ra g;
where pm = P (q m ). Further, 8 (x; y) 2 Mp denote
(
p
if x < y
ep (x; y) = pe (x; y)x
_
R
p
pe (y; x)(x ^ X) if x y,
ep (x; y) = Rp (x; y), 8 (x; y) 2
and R
= Mp .
We synthesis the …rst fact for the e¢ cient and proportional rules:
_
F1
(E) 8y 2 (0; X) and x
(P) 8y 2 (0; q m ) and x
r 1 (y), @x+ Re (x; y) = P 0 (x + y) x + P (x + y) ;
ep (x; y) = P 0 (x + y) x + P (x + y) :
q m y, @x+ R
This is shown in Lepore (2008): for r = e in the proof of Lemma 2, while for r = p, this
is fact (P1).
The next two facts are only relevant for proportional rationing and shown in Lepore
(2008) as (P2) and Theorem 2, respectfully.
F2
F3
ep (q m
8 (xo ; y) 2 Mp , y 2 [0; q m ], @x+ R
ep (x; y)
8 (x; y) 2 Mp , R
Rp (x; y) :
ep (xo ; y) :
y; y) > @x+ R
We also use the strict concavity of the Cournot revenue function, which we denote (C).
Proof of Theorem 2.
(Necessity) If Q is an equilibrium, then (y). Suppose to the
r
contrary that qa ; qa 2
= C and Q is an equilibrium.
De…ne Ar (qa ) = f
_
0
2 A j qa ; q
0
qa < X, it must be that (Ar (qa )) > 0.
2 Mr g. Based on A3 and since Q is such that
First take r = e, then @x+ Re qa ; q = 0 8 2 Ae (qa ), since the larger …rm’s capacity
plays no role in its expected revenue. In the Cournot game P 0 qa + q qa + P qa + q < 0
8 2 Ae (qa ).
Second take r = p, then @x+ Rp qa ; q
0 8 2 Ap (qa ), since the larger …rm’s capacity
always weakly increases its expected revenue.
8
Hence, 8 2 Ar (qa ),
@x+ Rr qa ; q
0 > P 0 qa + q
qa + P qa + q
:
For r 2 fe; pg, if 2 A r Ar (qa ), then from (F1) and (F3), @x+ Rr qa ; q = @x+ Rc qa ; q .
Hence, all expected marginal revenues are weakly less in either capacity choice game, and
for a positive probability mass of states the marginal revenues are strictly less. Putting this
together, @x+ ra (qa ; Q ) > @x ca (qa ; Q ) = 0, which contradicts qa 2 ra (Q ).
(Su¢ ciency) If (y), then Q is an equilibrium. Suppose to the contrary that 9a 2 A such
that qa 2
= ra (Q ) and (y). Since (y), ra (qa ; Q ) = ca (qa ; Q ), 8a 2 A. We use this to show
there cannot be a pro…t increasing defection from qa , 8a 2 A. We can immediately dismiss
qa < qa , because qa leads to Cournot pricing, 8a 2 A. Thus,
r
a (qa ; Q
)=
c
a (qa ; Q
)
where the weak inequality it follows from qa 2
c
a (qa ; Q
a (Q
)=
r
a (qa ; Q
);
).
We are left to show 8a 2 A defections qa > qa such that qa ; qa 2 Mr cannot be pro…t
increasing.
First take r = e, and there are two cases: (i) qa q and (ii) qa < q . (i) is immediate
since, @x+ Re (qa ; q ) = 0 < @x+ Re (qa ; q ) = P 0 (qa + q ) qa + P (qa + q ) and (ii) follows
immediately from fact (F1). Thus, 8a 2 A and qa > qa ,
@x+
e
a (qa ; Q
) < @x+
e
a (qa ; Q
) = @x
c
a (qa ; Q
) = 0;
a contradiction.
ep (q ; q ),
ep (qa ; q )
Next take r = p. We …rst show that for all such qa , @x+ R
@x+ R
a
p
8 2 A (qa ). This shown through the following sequences of inequalities, where the fact
used is in parentheses above the relation:
(F2)
@x+ Rp (qa ; q ) < @x+ Rp (q m
q ;q )
(F1)
= P 0 (q m ) (q m
(C)
q ) + P (q m )
P 0 (qa + q ) qa + P (qa + q ) .
er (qa ; q ) = @x P (qa ; q ). Putting
8 2 A r Ap (qa ), pricing is Cournot and from (F1), @x R
this all together, 8a 2 A and qa > qa ,
@x+ e pa (qa ; Q ) < @x+ e pa (qa ; Q ) = @x
c
a (qa ; Q
) = 0:
9
Thus, e pa (qa ; Q ) <
c
a (qa ; Q
): combined with (F3) this implies
p
a (qa ; Q
e p (qa ; Q ) <
a
)
a contradiction.
3.1
c
a (qa ; Q
);
Examples
The following examples provide a context for understanding condition (y).
Take the inverse demand P (q) = maxf
q; 0g, and uniform distributed uncertainty on
_
A. We characterize the set of lower and upper bounds marginal costs, a and a, such that
the there is an equilibrium of the capacity choice game with a UC outcome.
For e¢ cient rationing, (y) reduces to
_
a
(9)
3a.
The range of cost bounds such that Q is an equilibrium is shown in Figure 1.
For proportional rationing, (y) reduces to
_
a
3a
2
:
(10)
The range of cost bounds such that Q is an equilibrium is shown in Figure 2.
Thus, the greater the lowest bound marginal cost and the less the span between the
marginal cost bounds, the more likely the UC outcome is an equilibrium.
References
[1] Billingsley, P., 1986. Probability and Measure, John Wiley & Sons.
[2] Dasgupta, P. and Maskin, E., 1986. The Existence of Equilibrium in Discontinuous Economic Games, I: Theory. Review of Economic Studies 53, 1-26.
[3] Davidson, C. and Deneckere, R., 1986. Long-Run Competition in Capacity, Short-Run
Competition in Price, and the Cournot Model. RAND Journal of Economics 17, 404-415.
[4] Deneckere, R., and Kovenock, D., 1992. Price Leadership. Review of Economic Studies
59, 143-162.
[5] Kreps, D. M. and Scheinkman, J. A. 1983. Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes. Bell Journal of Economics 14, 326-337.
10
[6] Lepore, J. J., 2008. Consumer Rationing and the Cournot Outcome. Mimeo, California
Polytechnic State University, San Luis Obispo.
[7] Vives, X., 1999. Oligopoly Pricing: Old Ideas and New Tools, The MIT Press.