If duopolistic firms can choose their strategy variable, uncertainty about demand conditions and the degree of substitutability have countervailing effects on variable choice. High uncertainty favors prices, while close substitutability favors quantities. For intermediate values, a hybrid equilibrium exists.Competition, Strategy Variables, Demand Uncertainty

Ludwig Ressner

Markus Reisinger

English

Research Papers in Economics

 
 
 
 
 
 
BGPE Discussion Paper 
 
No. 7 
 
 
 
The Choice of Prices vs. Quantities under 
Uncertainty 
 
 
 
Markus Reisinger, Ludwig Ressner 
 
 
October 2006 
 
 
 
ISSN 1863-5733 
 
Editor: Prof. Regina T. Riphahn, Ph.D.   
Friedrich-Alexander-University Erlang n-Nuremberg  e
© Markus Reisinger, Ludwig Ressner 
 
 The Choice of Prices vs. Quantities under
Uncertainty∗
Markus Reisinger† and Ludwig Ressner‡
This Version: October 2006
Abstract
If duopolistic ﬁrms can choose their strategy variable, uncertainty
about demand conditions and the degree of substitutability have coun-
tervailing eﬀects on variable choice. High uncertainty favors prices,
while close substitutability favors quantities. For intermediate values,
a hybrid equilibrium exists.
Keywords: Competition, Strategy Variables, Demand Uncertainty
JEL Classiﬁcation: D43, L13
1 Introduction
If ﬁrms are free to choose their strategy variable, namely either prices or
quantities, it is well known that without uncertainty they prefer to choose
quantities if their products are substitutes. The reason is that quantities
soften competition and thereby guarantee higher equilibrium proﬁts com-
pared to prices. Singh and Vives (1984) show in a deterministic two-stage
game, in which ﬁrms ﬁrst choose their strategy variable and compete af-
terwards, that quantities are a dominant action. However, their analysis
∗We are very grateful to Richard Schmidtke and especially Sven Rady and Heiner
Schumacher for many helpful comments and suggestions.
†Corresponding Author: Department of Economics, University of Munich, Kaulbach-
str. 45, 80539 Munich, Germany, e-mail: markus.reisinger@lrz.uni-muenchen.de, phone:
00 49 89 2180 5645, fax: 00 49 89 2180 5650
‡Department of Economics, University of Munich, Kaulbachstr. 45, 80539 Munich,
Bavarian Graduate Program in Economics, Germany, e-mail: ludwig.ressner@lrz.uni-
muenchen.de
1abstracts from the fact that ﬁrms may face uncertainty at the time the
strategy variable has to be chosen. Our analysis incorporates this aspect by
introducing uncertainty via shocks that aﬀect the slope and the intercept of
the demand curve.1
In this framework the expected equilibrium proﬁt of a price setting ﬁrm
increases in the amount of uncertainty. However, expected equilibrium prof-
its decrease with product substitutability to a greater extent for a price set-
ting than for a quantity setting ﬁrm. As a result both ﬁrms select quantities
if uncertainty is small and prices if uncertainty is high relative to the degree
of substitutability. A “hybrid” equilibrium in which one ﬁrm chooses a price
and the other one a quantity emerges in an intermediate range.2
Our analysis relates to Klemperer and Meyer (1986). They allow for a
stochastic demand function and analyze which variable adapts more ﬂexibly
to shocks under diﬀerent cost functions. They consider a one-shot game and
therefore the comparative advantage of quantities is not present. Our anal-
ysis derives conditions under which either strategy variable’s comparative
advantage dominates.
In Section 2, we present the model, solve for the equilibrium, and provide
an intuition for our results. Section 3 concludes.
2 The Model
We consider a duopoly with diﬀerentiated products. Firms face the linear
demand system
pi = α −
β
θ
qi −
γ
θ
qj + ,
pj = α −
β
θ
qj −
γ
θ
qi + ,
with α > 0, and β ≥ γ ≥ 0. When γ = β, products are perfect substitutes
whereas with γ = 0 they are independent. θ and  are two random variables.
Without loss of generality we set E[θ] = 1, E[] = 0, and E[1/θ] = z.3 We
assume that the covariance between θ and  is nonnegative, σθ = ρσθσ ≥ 0.
So given a positive shock on the intercept ( > 0) the expected slope becomes
1For an analysis of a social planner facing demand uncertainty, see Weitzman (1974).
2This seems to be in line with empirical research. For example, Aiginger (1999) asked
managers of 930 manufacturing ﬁrm in Austria if they select prices or quantities as their
decision variable. Roughly, 2/3 charge prices and 1/3 set quantities.
3Since E[θ] = 1 we have by Jensen’s inequality that z > 1 if σθ > 0.
2ﬂatter (E[θ| > 0] > 1).4 To avoid unnecessary complications we require
the support of the shocks to be suﬃciently small such that no equilibria
emerge in which a price setting ﬁrm sells a negative quantity or a quantity
setting ﬁrm receives a negative price. Further we assume that ﬁrms have
zero costs.5
Competition between ﬁrms takes the form of a two-stage game. In stage 1
ﬁrms simultaneously choose their strategy variables. Each ﬁrm observes the
other ﬁrm’s choice and competes in stage 2 contingent on the chosen strategy
variables. Afterwards shocks realize, markets clear and proﬁts accrue. So
after the ﬁrst stage ﬁrms are committed to their strategy variable and cannot
change it thereafter. We solve for the subgame perfect equilibrium.
As spelled out before, it is a dominant strategy for ﬁrms to set quantities
in the ﬁrst stage of the deterministic game (σθ = σ = 0) since they induce
a lower degree of competition. Now turn to the case of uncertainty. We get
the following result:
Proposition The subgame perfect equilibrium outcome of the two stage
game is the following:
Both ﬁrms select a quantity in the ﬁrst stage if
max[0,σ?
θ] > σθ ≥ 0.
One ﬁrms selects a price and the other ﬁrm a quantity in the ﬁrst stage if
max[0,σ??
θ] > σθ ≥ max[0,σ?
θ].
Both ﬁrms select a price in the ﬁrst stage if
σθ ≥ max[0,σ??
θ] and γ < γ+.
If γ > γ+ there exists no equilibrium in which both ﬁrms select a price.
The values for σ?
θ, σ??
θ, and γ+ are deﬁned in (6), (8), and (9).
Proof We solve the game by backward induction. First, suppose that
both ﬁrms set prices as their strategy variable. Then in the second stage
4We restrict the covariance to be nonnegative for two reasons. Firstly, it seems natural
that if demand conditions are good both the intercept and the slope are hit by a positive
shock and vice versa if demand conditions are bad. Second, the analysis in case of negative
correlation would be very similar and is therefore omitted.
5As Singh and Vives (1984) show, the analysis would not change if ﬁrms faced positive
constant marginal costs c because this would only lower the eﬀective intercept from α to
a = α − c.
3each ﬁrm maximizes its expected proﬁt by choosing
max
pi
E
"
pi
 
θ((α + )(β − γ) − βpi + γpj)
β2 − γ2
!#
.
Since E[θ] = 1, E[] = 0, and E[θ] = σθ this is equivalent to
max
pi
pi
 
(α + σθ)(β − γ) − βpi + γpj
β2 − γ2
!
. (1)
Solving (1) for both ﬁrms yields equilibrium prices of pi = pj = p? =
(β−γ)(α+σθ)
2β−γ . Therefore each ﬁrm’s expected proﬁt is equal to
Π
p?p?
i =
(α + σθ)2(β − γ)β
(β + γ)(2β − γ)2 .
Next, suppose that both ﬁrms set quantities as their strategy variable. In
the second stage each ﬁrm maximizes its expected proﬁt by choosing
max
qi
E
"
qi
 
 + α −
βqi + γqj
θ
!#
. (2)
Since E[1
θ] = z the maximization problem in (2) is equivalent to
max
qi
qi(α − z(βqi + γqj)). (3)
Solving (3) for both ﬁrms yields equilibrium quantities of qi = qj = q? =
α
z(2β+γ) and an expected proﬁt of
Π
q?q?
i =
α2β
z(2β + γ)2 (4)
for each ﬁrm.
Lastly, if ﬁrm i chooses a price while ﬁrm j sets a quantity the equilibrium
price and quantity are p# =
(β−γ)(2z(β+γ)(α+σθ)−αγ)
4z(β2−γ2)+γ2 and q# =
α(2β−γ)+γσθ
4z(β2−γ2)+γ2.
The expected proﬁt of the price setting ﬁrm is
Π
p#q#
i =
(β − γ)2(2z(β + γ)(α + σθ) − αγ)2
(4z(β2 − γ2) + γ2)2β
,
while the expected proﬁt of the quantity setting ﬁrm is
Π
p#q#
j =
z(β2 − γ2)(α(2β − γ) + γσθ)2
(4z(β2 − γ2) + γ2)2β
.
4We now proceed to the ﬁrst stage. First look at the case in which ﬁrm j
sets a quantity. Firm i is indiﬀerent between setting a price or a quantity if
(β − γ)2(2z(β + γ)(α + σθ) − αγ)2
(4z(β2 − γ2) + γ2)2β
−
α2β
z(2β + γ)2 = 0. (5)
From (5) we derive the threshold covariance σ?
θ, above which ﬁrm i prefers
to charge a price. It is implicitly deﬁned by
σ?
θ ≡ α

z
γ
2(β − γ)
− z2 +
√
zβ(4z(β2 − γ2) + γ2))
2z2(β2 + γ2)(2β + γ)

.6 (6)
Note that an increase in σθ causes the left hand side of (6) to increase
whereas the right hand side decreases due to the positive relation between
σθ and z. Therefore σ?
θ is the relevant benchmark for covariances involving
the same σθ. The threshold covariance is positive if γ ≥ γ0, where γ0 is
implicitly deﬁned by
0 = f(γ0) ≡ (β+γ0)(2β+γ0)(zγ0−2z2(β−γ0))+
√
zβ(4z(β2−(γ0)2)+(γ0)2).
Since f(γ) is strictly convex in γ with f(0) < 0 and f(β) > 0, it follows that
γ0 ∈ (0,β) is unique. Thus for σθ ≥,σ?
θ ﬁrm i prefers to set a price.
Now suppose ﬁrm j sets a price. Then ﬁrm i is indiﬀerent between
choosing a price or a quantity if
(α + σθ)2(β − γ)β
(β + γ)(2β − γ)2 −
(β2 − γ2)z(α(2β − γ) + γσθ)2
(4z(β2 − γ2) + γ2)2β
= 0. (7)
From (7) we derive the threshold covariance σ??
θ.7 It is implicitly deﬁned by
σ??
θ ≡
α

γ(β + γ)(2β2(4β2 − 6βγ + γ2) + γ3(5β − γ))z − 16β2(β2 − γ2)2z2 − β2γ4

16β2(β2 − γ2)2z2 + γ2(β + γ)(4β3 − 8β2γ + 3βγ2 − γ3)z + β2γ4
+
2α
√
zβ(2β − γ)(β2 − γ2)(4z(β2 − γ2) + γ2)
16β2(β2 − γ2)2z2 + γ2(β + γ)(4β3 − 8β2γ + 3βγ2 − γ3)z + β2γ4.
(8)
6Since (5) is quadratic in σθ there exists a second threshold covariance which is strictly
negative and thus can be neglected.
7As in the previous case, there exists a second threshold covariance, that is strictly
negative and is therefore not relevant. Again an increase in σθ causes the left hand side
of (8) to increase and its right hand side to decrease.
5Now we determine the range in which σ??
θ ≥ 0. Let γ00 and γ+ be
implicitly deﬁned by
0 = g(γ00) ≡ γ00(γ00 + β)(β2(8β2 + 2(γ00)2 − 12γ00β) + (γ00)3(5β − γ00))z − β2(γ00)4
− 16β2(β2 − (γ00)2)2z2 + 2
√
zβ(2β − γ00)(β2 − (γ00)2)(4z(β2 − (γ00)2) + (γ00)2),
and
0 = h(γ+) ≡ 16β2(β2 − (γ+)2)2z2
+ (γ+)2(β + γ+)(4β3 − 8β2γ+ − 3β(γ+)2 − (γ+)3)z + (γ+)4β2,
(9)
where g(γ) is the numerator of (8) divided by α and h(γ) is the denominator
of (8).
First, we show, that γ00,γ+ ∈ (0,β) are unique. Since g(γ) decreases in
γ for γ smaller than some γ? ∈ (0,β) and strictly increases thereafter with
g(0) < 0 and g(β) > 0 it follows that γ00 ∈ (0,β) is unique. Similarly, h(γ)
increases in γ for γ smaller than some γ?? ∈ (0,β) and strictly decreases
thereafter with h(0) > 0 and g(β) < 0. Therefore γ+ ∈ (0,β) is unique.
Next we show that γ00 < γ+. Since σ??
θ is strictly convex in γ for γ ∈ [0,γ+)
with σ??
θ < 0 if γ = 0 and limγ→γ+ σ??
θ → ∞ it follows that γ00 < γ+. Thus,
the numerator and the denominator of σ??
θ have the same sign if and only if
γ ∈ (γ00,γ+). Therefore σ??
θ ≥ 0 if γ ∈ [γ00,γ+).
Now we show that for γ00 ≤ γ < γ+ ﬁrm i prefers to set a quantity
contingent on ﬁrm j choosing a price if 0 ≤ σθ < σ??
θ. Firm i sets a
quantity if the diﬀerence in expected proﬁts, given by the left hand side of
(7), is negative. Diﬀerentiating the left hand side of (7) twice with respect
to σθ yields that it is convex if
16β2(β2 −γ2)2z2 +γ2(γ +β)(4β3 −8β2γ +3βγ2 −γ3)z +γ4β2 = h(γ) > 0,
which is the case if γ < γ+. Thus, if γ00 ≤ γ < γ+, then for σθ < σ??
θ ﬁrm i
prefers to set a quantity, while for σθ ≥ σ??
θ it prefers a price.
Now consider the case γ > γ+. This implies that σ??
θ is negative and the
diﬀerence in expected proﬁts is concave in σθ. Consequently, ﬁrm i prefers
to set a quantity for every positive covariance.
Finally, it is easy to show that σ??
θ > σ?
θ for all γ00 ≤ γ < γ+. This
implies that γ0 > γ00. So for γ ≥ γ0 both ﬁrms select a quantity if 0 ≤
σθ < σ?
θ. For γ ≥ γ00 one ﬁrm selects a quantity and the other one a
price if max[0,σ?
θ] ≤ σθ < σ??
θ. Both ﬁrms select prices if γ < γ+ and
σθ ≥ max[0,σ??
θ]. 
6-
6
γ
σθ σ??
θ σ?
θ
γ+ γ00 γ0 β
(p,p)
(q,q)
(p,q)
Figure 1: Outcome of the game
The outcome of the game is depicted in Figure 1. One can see that
the degree of substitutability and the amount of uncertainty have oﬀsetting
eﬀects.
The intuition behind the result is the following: the closer substitutes
the two products are, the ﬁercer is competition but even more so under price
competition than under quantity competition. Thus, the higher γ, the more
ﬁrms are inclined to set quantities. If products are nearly perfect substitutes
(γ → β), then σ?
θ → ∞ and quantities are the preferred choice for every
positive covariance.
By contrast, the expected proﬁt of a price setting ﬁrm increases in every
component of σθ while that of a quantity setting ﬁrm decreases in σθ irre-
spective of the strategy variable chosen by the other ﬁrm. The underlying
reason is that if demand shifts upwards it also becomes ﬂatter in expec-
tation, while it becomes steeper with a downward shift. Therefore with
ﬁxed prices, the quantity increase in a good demand state overcompensates
in expectations the loss in a bad demand state. By contrast, if quantities
are ﬁxed the price drop in the bad state overcompensates in expectations
the price increase in the good state. So if γ ≤ γ00 we have that σ??
θ < 0.
Thus prices are preferred even if ρ and therefore σθ = 0 when demands are
suﬃciently independent.8
There also exists a hybrid equilibrium. Consider a γ ∈ (γ00,γ+). For
σθ = σ?
θ uncertainty is suﬃciently high such that ﬁrm i sets a price. Still
ﬁrm j prefers to set a quantity because this softens competition. As a
consequence, there always exists some intermediate range of σθ in which
ﬁrms play a hybrid equilibrium. Yet, in this equilibrium ﬁrm j’s proﬁt
8Therefore the uncertainty eﬀect is of particular importance in a monopolistic setting,
see Reis (2006).
7increases in the covariance by less than if it charged a price and so for
σθ ≥ σ??
θ the quantity setting ﬁrm leaves the hybrid equilibrium and ends
up setting a price. For γ ∈ [γ+,β], the products are such close substitutes
that even for an arbitrarily high covariance no (price, price) equilibrium
emerges.
3 Conclusion
This paper shows that the superiority of quantity competition for ﬁrms
might no longer hold if there is a substantial amount of uncertainty con-
cerning demand conditions. It also provides the testable implication that if
ﬁrms have some degree of choice about their strategy variable, they should
tend to choose quantities in industries with relatively stable and certain
demand, but choose prices if demand is ﬂuctuating and uncertain.
References
• K. Aiginger, 1999, The Use of Game Theoretical Models for Empirical
Industrial Organization, in: D.C. Mueller, A. Haid and J. Weigand,
eds., Competition, Eﬃciency, and Welfare - Essays in Honor of Man-
fred Neumann (Kluwer Academic Publishers, Netherlands), 253-277.
• Klemperer P. and M. Meyer, 1986, Price Competition vs. Quantity
Competition: The Role of Uncertainty, RAND Journal of Economics
17, 618-638.
• R. Reis, 2006, Inattentive Producers, Review of Economic Studies 73,
793-821.
• Singh N. and X. Vives, 1984, Price and Quantity Competition in a
Diﬀerentiated Duopoly, RAND Journal of Economics 15, 546-554.
• M. Weitzman, 1974, Prices vs. Quantities, Review of Economic Stud-
ies 41, 477-491.
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