collusive price rigidity under price- matching...
TRANSCRIPT
ISSN 1745-9648
Collusive Price Rigidity under Price-
Matching Punishments
Luke Garrod School of Business and Economics, Loughborough
University
CCP Working Paper 11-14
Abstract: In this paper we provide game theoretic support for the results of the kinked demand curve. By analysing an infinitely repeated game where unit costs fluctuate stochastically between a low and a high state over time and where firms follow a price-matching punishment strategy, we demonstrate that price rigidity can occur in the best collusive subgame perfect Nash equilibrium for small fluctuations in costs. The critical level of high costs under which the best collusive prices are rigid is shown to depend upon the expected duration of a sequence of high-cost periods, the number of firms in the market, and the degree of product differentiation. September 2011
JEL Codes: L11; L13; L41 Keywords: Tacit collusion; kinked demand curve; price rigidity. Acknowledgements: I am grateful for comments from Iwan Bos, Steve Davies, Joe Harrington, Morten Hviid, Roman Inderst, Kai-Uwe Kühn, Bruce Lyons, Patrick Rey, Paul Seabright, Christopher M. Wilson, and seminar participants at the International Industrial Organization Conference 2011 and the European Association of Research in Industrial Economics Conference 2011. The support of the Economic and Social Research Council (UK) is gratefully acknowledged. The usual disclaimer applies.
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Contact Details: Luke Garrod, School of Business and Economics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom, email: [email protected]
Collusive Price Rigidity under Price-Matching
Punishments�
Luke Garrody
September 2011
Abstract
In this paper we provide game theoretic support for the results of the kinked
demand curve. By analysing an in�nitely repeated game where unit costs �uc-
tuate stochastically between a low and a high state over time and where �rms
follow a price-matching punishment strategy, we demonstrate that price rigidity
can occur in the best collusive subgame perfect Nash equilibrium for small �uc-
tuations in costs. The critical level of high costs under which the best collusive
prices are rigid is shown to depend upon the expected duration of a sequence of
high-cost periods, the number of �rms in the market, and the degree of product
di¤erentiation.
Keywords: Tacit collusion, kinked demand curve, price rigidity
JEL: L11, L13, L41
1 Introduction
There has been a long-standing belief in industrial economics that tacit collusion
and price rigidity are linked. This belief was �rst formalised by the old theory of the
kinked demand curve (Hall and Hitch, 1939; and Sweezy, 1939). This theory assumes
that a �rm expects its rivals will continue to set a certain focal price if it charges
more than this focal level, but expects its rivals will match any price set below this
focal level. Consequently, the �rm�s demand curve is kinked at the focal price, and
the resultant discontinuity in its marginal revenue curve implies that prices remain
at the focal level for small cost �uctuations. Although the assumed rivalry has an
�I am grateful for comments from Iwan Bos, Steve Davies, Joe Harrington, Morten Hviid, RomanInderst, Kai-Uwe Kühn, Bruce Lyons, Patrick Rey, Paul Seabright, Christopher M. Wilson, andseminar participants at the International Industrial Organization Conference 2011 and the EuropeanAssociation of Research in Industrial Economics Conference 2011. The support of the Economicand Social Research Council (UK) is gratefully acknowledged. The usual disclaimer applies.
ySchool of Business and Economics, Loughborough University, Loughborough, Leicestershire,LE11 3TU, United Kingdom, email: [email protected]
1
intuitive appeal and some anecdotal support, the kinked demand curve has been
heavily criticised (for example see Tirole, 1988, p.243-245).
More recently, the way in which dynamic oligopolistic interaction has been mod-
elled di¤ers in two respects with the kinked demand curve. First, it is modelled as
an explicit dynamic game using the theory of repeated games, where collusive prices
are sustainable when the short-term bene�t from any deviation is outweighed by
a credible and su¢ ciently harsh long-term retaliation. Second, �rms usually more
than match lower deviation prices, because the most commonly analysed retaliations
are �Nash reversion�(see Friedman, 1971) and �optimal punishment strategies�(see
Abreu, 1986, 1988). Since market conditions can a¤ect the deviant�s gains and its
rivals�retaliation, there is a theoretical literature that analyses how market �uctua-
tions a¤ect the best collusive prices that achieve the highest levels of pro�t possible
(for example see Rotemberg and Saloner, 1986; Haltiwanger and Harrington, 1991).1
One feature of this literature is that, barring the special circumstances when incen-
tives are perfectly aligned, the best collusive prices vary with �uctuations of any
size. This is in contrast with the belief that tacit collusion and price rigidity are
linked, and it is at odds with the results of the kinked demand curve.
In contrast to the previous repeated game literature, in this paper we provide
game theoretic support for the results of the kinked demand curve by showing that
the best collusive prices can be rigid in an in�nitely repeated game for small industry-
wide cost �uctuations. We derive this result by extending Lu and Wright (2010)
who analyse an in�nitely repeated game under price-matching punishments where,
similar to the rivalry of the kinked demand curve, �rms match lower deviation
prices (provided they are above the one-shot Nash price). They show that price-
matching punishments can support collusive prices when products are symmetrically
di¤erentiated and market conditions do not vary over time. We extend their model
so that unit costs �uctuate stochastically between a high and a low state over time,
and show that the best collusive prices can be rigid over time.
The intuition is that when costs are temporarily high today but are permanently
low in the future, for any price today that is above the future�s price there is a
unilateral incentive to set the future�s price today (provided today�s one-shot Nash
price is below the future�s price). This is because under price-matching punishments
the future�s price is set in the future whether there is such a deviation today or not.
When high costs persist into the future, price matching such a deviation does reduce
pro�ts in future high-cost periods, and the pro�tability of such a deviation decreases
as the high-cost level increases. As a result, procyclical prices are supportable when
cost �uctuations are su¢ ciently large. Since an increased likelihood of future high-
1Although much of the previous literature focuses on demand �uctuations, many of the resultsgeneralise to cost �uctuations.
2
cost periods also makes such a deviation less pro�table, the critical level of high
costs under which the best collusive prices are rigid is monotonically decreasing in
the expected duration of a sequence of high-cost periods, and prices are not rigid
when costs are permanently high in the future.
Given our model is based on game theoretic foundations, it generates richer
predictions than the kinked demand curve regarding how prices and pro�ts respond
to the other parameters of the model. There exists a unique price that achieves the
highest level of pro�t given the constraint that the price does not vary with costs.
This price de�nes the best collusive price in both cost states when �uctuations are
small, and it is always between the high-cost one-shot Nash price and the low-cost
monopoly level under such conditions. It monotonically increases with the level of
high costs, but only a small proportion of the high costs are passed through via a
higher price. Consequently, as �uctuations become more pronounced, low-cost per-
period pro�ts increase but high-cost per-period pro�ts decrease. When considered
over time, this increase in pro�t is outweighed by the decrease, so each �rm�s present
discounted value of collusive pro�ts is decreasing in the level of high costs despite
the higher price.
Our model also generates predictions regarding the relationship between price
rigidity and the number of �rms in the market, which has been investigated by several
empirical studies (for example see Carlton, 1986, 1989). This relationship ultimately
depends upon the degree of product di¤erentiation. Based on an example where
demand is derived from the constant elasticity of substitution version of Spence-
Dixit-Stiglitz preferences (Spence, 1976; and Dixit and Stiglitz, 1977), we show
that the best collusive prices are rigid for reasonably large cost �uctuations when
products are di¤erentiated by a intermediate degree. This is because price-matching
punishments do not support collusive prices when products are homogeneous and
since �rms are unconcerned by rivals�responses when they are local monopolies. This
example also shows that the best collusive prices are rigid for larger cost �uctuations
as the number of �rms decreases when products are su¢ ciently substitutable.
The rest of the paper is structured as follows. Section 2 reviews the related
literature and provides anecdotal support for a link between price matching and
price rigidity. Section 3 outlines the assumptions on demand and costs, and it
formally de�nes the price-matching punishment strategy. In section 4 we derive the
conditions where the best collusive prices are rigid, and investigate the relationship
between price rigidity and the expected duration of a sequence of high-cost periods,
as well as considering the e¤ects of cost �uctuations and price rigidity on the best
collusive pro�ts. Section 5 places more structure on demand to investigate how price
rigidity relates to both the number of �rms in the market and the degree of product
di¤erentiation, and section 6 concludes. All proofs are relegated to the appendix.
3
2 Related Literature and Evidence
In this paper we propose that the expectation that deviation prices will be matched
can lead to price rigidity during collusive phases, and there is some anecdotal support
for this. Slade (1987, 1992) analysed a price war between gasoline retailers during
1983 in Vancouver (see also Slade, 1990). During the price war, she found there was
�a high degree of (lagged) price matching�and �prices before and after the war were
uniform across �rms and stable over time� (1992, p.264). In fact, �after the price
war came to an end, prices were stable for nearly a year�(1987, p.515). Slade (1989,
p.295) also argues that other Canadian markets (including nickel and cigarette, as
well as gasoline) had three stylised facts: �First, price is the choice variable and it
can be observed by all. Second, price wars are occasional events and are separated
by periods of stable prices. Third, during a war there is considerable matching of
prices�. Similarly, Kalai and Sattherwaite (1994) state that between 1900 and 1958
small �rms in the US steel industry believed the largest producer would match their
prices if they undercut it, and observed that �Before World War II certain classes
of steel products showed remarkable price rigidity�(p.31).2
Although informal reasoning suggests that �rms would prefer the harshest pun-
ishment, since it makes collusion easier to sustain, the above evidence suggests that,
in at least some situations, price matching is a relevant form of �rm behaviour. Of
particular importance in this regard is Slade�s (1987) empirical evidence that �nds
some support for punishment strategies, similar to price matching, where �small
deviations lead to small punishments�over Nash reversion (p.499). The above evi-
dence also suggests that our price rigidity result may be of some empirical relevance
for such situations where price matching is prevalent. This contrasts with previous
attempts to model the kinked demand curve in dynamic settings, because they do
not �nd a link between price matching and price rigidity.3
Our model also contrasts with the literature that analyses collusion under market
�uctuations. Rotemberg and Saloner (1986) show that collusion under Nash rever-
sion is harder to support in a period of boom than bust when future �uctuations
are independently and identically distributed. This is because the deviant�s gains
2Levenstein (1997) and Genesove and Mullin (2001) also �nd that some cartel price wars consistedof mild punishments and price matching, respectively, but due to infrequent price observations it isnot possible to determine the extent to which prices �uctuate over time.
3Bhaskar (1988) and Kalai and Satterthwaite (1994) show that price rigidity does not occur ina one-shot game when lower prices can be matched immediately before pro�ts are realised. In anin�nitely repeated game where a dupoly alternates between commiting to price for two periods,Maskin and Tirole (1988) argue that price rigidity can occur to avoid a price war when costs fallpermanently. However, this is because rivals more than match lower prices. In another relatedin�nitely repeated game, Slade (1989) captures the three stylised facts discussed above when anunexpected change in demand is anticipated to be permanent, but stable prices only occur in hermodel when the new equilibrium is reached.
4
are largest in a boom but the punishment remains the same. This implies that the
best collusive prices move procyclically with costs, because any price that is just
supportable in a high-cost period is not be supportable in a low-cost period. Due
to a similar reason, the incentive to deviate from a rigid price under price-matching
punishments is greatest in low-cost periods when future �uctuations are independent
of or positively correlated with the current level. However, this incentive does not
always translate into procyclical prices. This is because deviating from a procyclical
price in a high-cost period by setting the low-cost price limits the punishment a
�rm receives to future high-cost periods. Therefore, the deviant�s gains for a small
deviation from a price only slightly above the low-cost price is e¤ectively the same
as that for a small deviation from the low-cost price, but the punishment can be
much weaker. As a result, the low-cost price is supportable in a high-cost period,
but a price only slightly above the low-cost price may not be.
Finally, this paper is also related to Athey et al (2004) who develop an alter-
native model of collusive price rigidity where prices are publically observable but
�rms experience privately observed shocks to unit costs in each period. They show
that under Nash reversion the best collusive prices may be rigid over time because,
although demand is not allocated to the most e¢ cient �rm, this ine¢ ciency can be
outweighed by the bene�t of detecting deviations easily.4 Their model is similar
to Green and Porter (1984) since, due to some information asymmetry, price wars
occur on the equilibrium path when �rms receive a bad signal. In contrast, there is
symmetric information in our model, so price wars do not occur on the equilibrium
path. Instead, the successfulness of collusion is a¤ected by market conditions in a
similar way as Rotemberg and Saloner (1986). Our model adds to our understand-
ing of price rigidity because it is the �rst (to the author�s knowledge) to consider
the relationship between price rigidity and the degree of product di¤erentiation,
and given it does not rely on parameters that are likely to be unobservable to an
econometrician, it can be tested empirically.
3 The Model
3.1 Basic assumptions
Consider a market where a �xed number of n � 2 �rms each produce a single
di¤erentiated product and compete in observable prices over an in�nite number of
periods. In any period t, �rms have identical unit costs, ct � 0, face no �xed costs,and have a common discount factor, � 2 (0; 1). They simultaneously choose price in
4 In a similar model, Hanazono and Yang (2007) show that price rigidity can also occur withunobservable demand �uctuations.
5
each period and the demand of �rm i = 1; : : : ; n in period t is qi(pit;p�it; n) where
pit is its own price and p�it is the vector of its rivals�prices. Demand is symmetric,
strictly decreasing in pit and limpit!1 qi(pit;p�it; n) = 0. Since �rms are symmetric,
at equal prices pit = pt for all i, qi(pt; pt; n) = q(pt)=n where q(pt) is independent of
n. For every price vector pt=(pit;p�it) where qi(pit;p�it; n) > 0 for all i, demand
is twice continuously di¤erentiable and from Vives (2001, p.148-152) we assume it
has the following standard properties:
Assumption 1.��� @qi@pit
��� >Pj 6=i@qi@pjt
> 0
Assumption 2. @2qi@pit@pjt
� 0 8 j 6= i
Assumption 3. @2qi@p2it
+Pj 6=i
@2qi@pit@pjt
< 0.
These assumptions imply that products are imperfect substitutes, demand ex-
hibits increasing di¤erences in (pit; pjt) and the own e¤ect of a price change domi-
nates the cross e¤ect both in terms of the level and slope of demand.
Firm i�s per-period pro�t in period t is �it(pit;p�it; ct; n) = (pit�ct)qi(pit;p�it; n)where at equal prices pit = pt for all i write �it(pt; pt; ct; n) = �t(pt; ct; n). Assump-
tions 1 and 2 imply that prices are strategic complements:
@2�it@pit@pjt
> 0 8 j 6= i 8 t. (1)
Since unit costs are constant, Assumptions 1 and 3 are su¢ cient to ensure the best
reply mapping is a contraction (see Vives, 2001, p.150):
@2�it@p2it
+Xj 6=i
@2�it@pit@pjt
< 0 8 t. (2)
This guarantees the existence of a unique one-shot Nash equilibrium in pure strate-
gies, denoted pN (ct; n). It follows from (1) and (2) that each �rm�s pro�t is strictly
concave in its own price (i.e. @2�it=@p2it < 0), which implies that if rivals charge
above pN (ct; n), then a �rm can strictly increase its pro�t by unilaterally lowering
its price towards the one-shot Nash price (i.e. @�it=@pit < 0 8 pjt > pN (ct; n) j 6= i).Assumption 1 guarantees that pN (ct; n) is strictly increasing in ct and to ensure that
pN (ct; n) is strictly decreasing in n we assume the following su¢ cient condition:
Assumption 4. @2qi@pit@n
< 0.
6
Finally, to ensure that the monopoly price, pm(ct), is unique with pm(ct) >
pN (ct; n) we assume:
Assumption 5. d2�tdp2t
= @2�it@p2it
+ 2Pj 6=i
@2�it@pit@pjt
+ @2�it@p2jt
< 0 8 t.
An implication of Assumption 5 is that if all �rms set the same price below the
monopoly level, then they would strictly increase pro�ts if all set a higher price
(i.e. d�t=dpt > 0 8 pN (ct; n) � pt < pm(ct)). Assumption 1 ensures that pm(ct) isstrictly increasing in ct, while symmetric demand and costs guarantee that pm(ct)
is independent of n.
3.2 Cost �uctuations
In any period, unit costs can be low or high such that ct = 0 or ct = c > 0. To
simplify notation, write pN (0; n) = pN (n), pm(0) = pm and �it(pit;p�it; 0; n) =
�it(pit;p�it;n). The current level is common knowledge before �rms set prices, and
expectations of future levels of ct for all t follow a Markov process such that:
� � Pr (ct = cj ct�1 = 0) 2 (0; 1)� � Pr (ct = 0j ct�1 = c) 2 (0; 1)� � Pr (c0 = c) 2 [0; 1].
Thus, � is the transition probability associated with moving from a low-cost period
to one of high costs, and � is the probability that corresponds with a transition from
high costs to low costs. The parameter � describes how the system begins.
This process implies that the probability that costs will be high in the next
period is � if they are currently low, otherwise it is 1� �. Thus, when 1� ��� = 0,future cost are independent of the current level, and this simple case provides a
benchmark for our analysis. In many industries it is natural to expect future costs
will be positively correlated with the current level. Consequently, we also allow for
the case where 1 � � � � > 0, which implies that it is more likely that the currentlevel will continue into the next period than change. Following the terminology of
Bagwell and Staiger (1997), we refer to the former as zero correlation (1���� = 0)and the latter as positive correlation (1� � � � > 0).
3.3 Collusive prices and pro�ts
Due to the Markov process that determines future cost levels, collusive pro�ts are
the same in any high-cost period regardless of the speci�c date, other things equal,
and likewise for any low-cost period. Thus, the best collusive prices emerge as a pair,
7
and we wish to �nd the conditions under which these are equal. Analysing the best
collusive prices is consistent with the prominent papers in the collusion literature
(for example see Rotemberg and Saloner, 1986; Haltiwanger and Harringtion, 1991),
and it is also consistent with the kinked demand curve since the most pro�table
equilibrium is often argued to be the most logical (see Tirole, 1988, p.244).
To derive the present discounted values of collusive pro�ts, denote H(p(c); p(0))
as a �rm�s pro�t in period t and thereafter if period t has high costs and �rms
set p(c) and p(0) in all high- and low-cost periods, respectively. Similarly, denote
L(p(0); p(c)) as a �rm�s pro�t in period t and thereafter if period t has low costs and
�rms set p(0) and p(c) in all low- and high-cost periods, respectively. Surpressing
notation slightly, it is possible to write such pro�ts as:
H = �(p(c); c; n) + ��L + �(1� �)HL = �(p(0);n) + ��H + �(1� �)L.
Solving for H and L enables us to write �rm i�s present discounted value of
collusive pro�ts in a high-cost period and a low-cost period as, respectively:
H(p(c); p(0)) = �(p(c); c; n) +�1��
��!�(p(0);n) + (1�
�! )�(p(c); c; n)
�L(p(0); p(c)) = �(p(0);n) +
�1�� [
�!�(p(c); c; n) + (1�
�! )�(p(0);n)],
where ! � 1� �(1� � � �) > 0, 0 < �! < 1 and 0 <
�! < 1. The �rst terms on the
right hand-side of the above equations represent the pro�ts from the initial periods,
and the second terms represent the pro�ts from all future periods conditional on
expectations of future cost levels.
3.4 Punishment strategy
Drawing on the insights of Lu and Wright (2010), we assume that �rm i�s price-
matching punishment strategy pro�le for all t is of the form:
pi0 = p0(c0) = p(c0)
pit = pt(ct) =
(p(ct) if pj� = p� (c� ) 8 j 8 � 2 f0; : : : ; t� 1gmax(pN (ct; n);min(p
dt ; p(ct))) otherwise
(3)
where pdt is a vector of the history of deviation prices at period t (i.e. it includes all
prices where pj� 6= p� (c� ) 8 j 8 � 2 f0; : : : ; t� 1g). This strategy calls for each �rmto set the initial collusive prices until a deviation. Following a deviation, the lowest
ever deviation price is matched in periods where it is above the period�s one-shot
8
price
time
pN(c,n)
t+1
Z
pN(n)
Z
YY
X
pt(c)
t+2 t+3 t+4 t+5 t+6t
X XX
Y Y Y Y Y
Z Z
pt(0)
Figure 1: pricing after a one-stage deviation to X, Y and Z in period t
Nash price and below the period�s initial collusive level. The one-shot Nash price is
set in any period when the lowest ever deviation price is below the one-shot Nash
price. Similarly, the initial collusive level is set in any period when the lowest ever
deviation price is above the period�s initial collusive level. This is repeated for future
deviations.
Figure 1 illustrates the implications for pricing for various one-stage deviations
(i.e. where a �rm deviates for one period, then conforms to (3) thereafter). Un-
derstanding such deviations are important for our purposes, because we use the
one-stage deviation principle (see Fudenberg and Tirole, 1991, p.108-110) to solve
for subgame perfect Nash equilibria. This principle states that a strategy pro�le in-
duces an equilibrium in every subgame if there is no history that leads to a subgame
in which a player will chose an action that di¤ers to that prescribed by the strategy,
then conforms to the strategy thereafter (assuming the deviant believes others will
also conform to the strategy). Thus, to prove subgame perfection, it su¢ ces to show
that a �rm will not deviate once from the initial collusive subgame and nor will it do
so from every possible punishment subgame (where deviating once is synonymous
with a one-stage deviation). We say that p(ct) is supportable if the strategy pro�le
in (3) is a subgame perfect Nash equilibrium for all i = 1; : : : ; n.
In the particular example of Figure 1, �rms set pt(c) and pt(0) in high- and low-
cost periods, respectively, if there is no deviation over the period, so intitially prices
are procyclical. If a �rm deviates once in period t to Y , then Y is matched in all
future periods. If the deviation price is Z, however, then Z is matched in all future
low-cost periods but pN (c; n) is set in all others. Departing slightly from the kinked
demand curve but consistent with Lu and Wright (2010), �rms do not match prices
9
below pN (ct; n) in period t because doing so seems unreasonable. This assumption
is not crucial in determining when the best collusive prices are rigid, because a
deviation to Z in a low-cost period is always less pro�table than one to Y when the
best collusive prices are rigid, and a deviation to Z in a high-cost period never occurs
even for prices that are unsupportable by (3). This assumption ensures that it is
possible to check that any price between pN (n) and pN (c; n) de�nes an equilibrium
in such low-cost punishment subgames, and that (3) induces an equilibrium in a
subgame for histories where the price is below pN (ct; n).
The strategy pro�le (3) also has a similar feature for one-stage deviations to
prices above the lowest initial collusive price, because if the deviation price is X,
then X is matched in future high-cost periods, but pt(0) is set in all others. Figure
1 illustrates the case for procyclical prices, but it equally applies to the case of
countercyclical prices (where pt(c) is set in low-cost periods and pt(0) in high-cost).
This resembles the rivalry of the kinked demand curve where �rms are unwilling
to match a price increase above the focal level. The only di¤erence in the rivalry
modelled here is that this unwillingness to match prices above the initial collusive
level occurs regardless of whether a �rm deviates up from pt(0) to X or down from
pt(c) to X. However, the rationale for the strategy is the same: each �rm expects
to lose sales if it were to set X in periods when it expects its rivals to set pt(0).5
4 Price Rigidity under Price-Matching Punishments
4.1 A Theory of Price Rigidity
We wish to �nd the conditions under which the best collusive prices are rigid.
Prices are procyclical for large cost �uctuations where pN (c; n) > pm, so we ini-
tially consider cost �uctuations such that c 2 (0; c] where pN (c; n) = pm. Similarly,price rigidity does not occur unless �rms can support pN (c; n) and p, such that
pN (n) < p � pN (c; n), in high- and low-cost periods, respectively. Otherwise, thereare some punishment subgames where �rms will not conform to (3). Clearly a �rm
will not deviate once from pN (c; n) in a high-cost period, so consider �rm i�s incen-
tive to deviate once from p in a low-cost period. Firm i�s present discounted value
of deviation pro�t if it sets the same or a lower price pi 2 (pN (n); p] in a low-cost5An alternative strategy is one where downward deviations from pt(c) to X are matched in all
future periods, other things equal. Since this alternative and (3) are equivalent for rigid prices, thecharacteristics of the best rigid price are the same. We focus on (3) because there is an asymmetryin this alternative strategy since �rms are unable to lead low-cost prices up from pt(0) to X in alow-cost period, but they are able to do so by a downward deviation from pt(c) to X in a high-costperiod. A consequence of this asymmetry is that the parameter space where the best collusive pricesare rigid under (3) is a strict subset of that under this alternative strategy, so it is robust to both.
10
period and then conforms to (3) thereafter is:
ziL(pi; p; pN (c; n)) � �i(pi; p;n) + �
1�� [�!�(p
N (c; n); c; n) + (1� �! )�(pi;n)]. (4)
Lemma 1 For every n � 2 and � 2 (0; 1) where 1� ��� � 0, there exists a uniquebc 2 (0; c) such that pN (c; n) and p, where pN (n) < p � pN (c; n), are supportable by(3) in high- and low-cost periods, respectively, if and only if c 2 (0;bc].
When cost �uctuations are su¢ ciently small, pN (c; n) is close enough to pN (n)
such that the punishment from matching any lower price in future low-cost periods
ensures a �rm will not deviate once from pN (c; n) in a low-cost period. The pun-
ishment is also credible since a �rm will not deviate once from any lower price p
between pN (n) and pN (c; n) in low-cost punishment subgames. This is because the
condition for a �rm to want to deviate once from p is the same as that of pN (c; n)
except that p is lower than pN (c; n), and the standard properties of the underlying
competition game imply that it is less pro�table to deviate once from a price close
to pN (n) than a higher price.
In the next subsection we limit our attention to equilibria with the same price
pc > pN (c; n) in both cost states. This allows us to characterise the best rigid price
that achieves the highest level of pro�t given that the price does not vary with costs.
In the subsection after, we �nd the conditions under which �rms can do no better
than set the best rigid price in both cost states.
4.1.1 Best rigid price
Under the conditions of Lemma 1, price rigidity can occur when �rms can support
a rigid price pc > pN (c; n) and any rigid price between pc and pN (c; n). Otherwise
there is at least one collusive/punishment subgame where a �rm will not conform
to (3). Since a �rm�s incentives to deviate depend upon the current cost level, we
must ensure that a �rm will not deviate once from such prices in high- and low-cost
periods. Depending upon whether costs are initially high or low, �rm i�s present
discounted value of deviation pro�ts if it sets the same or lower price p 2 [pN (c; n); pc]in the initial period and then conforms to (3) thereafter are, respectively:
yiH(p; pc) � �i(p; pc; c; n) + �
1����!�(p;n) + (1�
�! )�(p; c; n)
�(5)
yiL(p; pc) � �i(p; pc;n) + �
1�� [�!�(p; c; n) + (1�
�! )�(p;n)]. (6)
The �rst terms on the right-hand side of (5) and (6) are the pro�t from the deviation
period. This pro�t is lower in (5) than (6), because per-period pro�ts are decreasing
in unit costs. The second terms represent the pro�t from future periods conditional
11
on expectations of future cost levels and given a lower price will be matched in all
future periods. If there is zero correlation, future �uctuations in costs are indepen-
dent of the current level, so the second terms of (5) and (6) are equal. If there is
positive correlation, the current cost state is more likely to continue into the future,
so the second term is lower in (5) than (6). Thus, such a one-stage deviation is less
pro�table in a high-cost period than a low-cost period, other things equal.
Lemma 2 For every n � 2 and � 2 (0; 1) where 1� ��� � 0, there exists a uniquebest rigid price, pyL(c; n; �; �; �), that is supportable by (3) if and only if c 2 (0;bc],where pN (c; n) < pyL(c; n; �; �; �) < pm(c). Any rigid price pc such that pN (c; n) �pc � pyL(c; n; �; �; �) is also supportable by (3).
When costs are small, there is some rigid price pc above pN (c; n) where the
punishment from matching a small deviation in all future periods ensures a �rm
will not deviate once from pc in a low-cost collusive subgame. Given a one-stage
deviation is most pro�table when costs are low, a �rm will also not deviate once from
pc in a high-cost collusive subgame. The punishment is credible since a �rm will
also not deviate once from any lower rigid price p between pN (c; n) and pc in low-
and high-cost punishment subgames. A �rm will not deviate once from p in low-cost
punishment subgames, because the condition for a �rm to want to deviate once from
p is the same as that of pc except that p is lower than pc, and the standard properties
of the underlying competition game imply that it is less pro�table to deviate once
from a price close to pN (c; n) than a higher price.6 This also ensures a �rm will not
deviate once from p in high-cost punishment subgames, due to the same reason as
for the initial collusive subgame.
The best rigid price has the property that its associated (unconstrained) optimal
�deviation�price in a low-cost period equals the best rigid price (i.e. the argument
maximising (6) equals pc). Since it is less pro�table to deviate once in a high-cost
period than one of low costs, its (constrained) optimal �deviation�price in a high-
cost period also equals the best rigid price.7 The best rigid price is equivalent to the
best collusive price analysed by Lu and Wright (2010) when costs do not �uctuate,
but in contrast to Lu and Wright (2010) it can be equal to or above the low-cost
monopoly level when costs do �uctuate over time. This is because a small deviation
from pm that is matched in all future periods generates a �rst-order decrease in
6Under the conditions of Lemma 1, a one-stage deviation from any rigid price p > pN (c; n) to aprice below pN (c; n) in a low-cost period is less pro�table than one to pN (c; n).
7A �rm would want to deviate once to a price above the best rigid price in a high-cost periodif such a deviation price were matched in all future periods. However, this would not be a crediblestrategy even if upward deviations were matched, because a �rm will deviate once from rigid pricesabove the best rigid price in such future low-cost punishment subgames.
12
future high-cost periods only (since the low-cost pro�t function is �at at pm), and
this can outweigh the �rst-order increase in the initial deviation pro�t.
4.1.2 Best collusive prices and price rigidity
The best collusive prices are rigid if �rms can do no better than the best rigid price
by setting procyclical or countercyclical prices. The best collusive prices are never
countercyclical, because the condition for a �rm to want to deviate once from a price
above the best rigid price in a low-cost period is the same for countercyclical prices
as it is for rigid prices except that for countercyclical prices a small deviation is only
matched in future low-cost periods. Thus, since there is an incentive to deviate once
from such a price when the price is matched in all future periods, there is also an
incentive to deviate once when prices are countercyclical. Therefore, �rms can only
do better than rigid prices by setting procyclical prices.
The best collusive prices are procyclical when �rms can support p(c) in a high-
cost period that is above the best collusive price in a low-cost period, denoted
p�(0) 2 [pN (c; n); pm]. Otherwise a �rm will deviate once from procyclical prices in
a high-cost period, and the best collusive prices are rigid. The pro�t from a small
one-stage deviation from p�(0) in a low-cost period is equivalent to (6), so consider
�rm i�s present discounted value of deviation pro�ts in a high-cost period if it sets
the same or a lower price p 2 [p�(0); p(c)] in the initial period, then conforms to (3)thereafter:
xiH(p; p(c); p�(0)) � �i(p; p(c); c; n) + �
1����!�(p
�(0);n) + (1� �! )�(p; c; n)
�.
(7)
To see why procyclical prices may not be supportable under price-matching punish-
ments, consider a one-stage deviation from p(c) to p = p�(0). When such a deviation
is matched in future periods, the punishment is limited to future periods of high-
costs (because �rms set p�(0) in low-cost periods regardless of a deviation). When
�rms expect all future periods to be low-cost (i.e. � = 1 and � = 0), there is no
punishment for such a deviation. Thus, each �rm will have a dominant strategy to
deviate once from any p(c) 2 (p�(0); pm(c)] for all c 2 (0;bc], and the best rigid priceis the best collusive price for both cost states.
Proposition 1 shows the best collusive price can be rigid when costs persist into
the future.
Proposition 1 For every n � 2 and � 2 (0; 1) where 1 � �� � � 0, there exists aunique c� 2 (0;bc) such that the best rigid price, pyL(c; n; �; �; �), is the best collusiveprice in both cost states if and only if c 2 (0; c�], where pN (c; n) < pyL(c; n; �; �; �) <pm.
13
Under the conditions of Lemma 1, the best collusive price in low-cost periods
is the lower of the best rigid price and the low-cost monopoly price regardless of
whether prices are rigid or procyclical. This is because for any p(0) above the
best rigid price there are some low-cost punishment subgames where a rigid price
between pyL(c; n; �; �; �) and p(0) should be matched in all future periods, but a �rm
will deviate once. Consequently, prices above the best rigid price are not supportable
even though it is easier to support the best rigid price in low-cost periods when prices
are procyclical than when they are rigid.
The best collusive prices are rigid when a �rm has an incentive to deviate once
from a price slightly above the best rigid price in a high-cost period. This occurs
for small cost �uctuations, because as c ! 0 the condition for a �rm to want to
deviate once from any price above the best rigid price in a high-cost period is the
same as that of a low-cost period except that a small deviation is only matched in
high-cost periods. Thus, since there is an incentive to deviate once from such prices
in a low-cost period when a small deviation is matched in all future periods, there is
also an incentive to deviate once when costs are high. As c increases, it becomes less
pro�table to deviate once from a price slightly above the best rigid price in high-cost
periods (even though this price strictly increases with c), and procyclical prices are
supportable by (3) when cost �uctuations are su¢ ciently large.
When �uctuations are so large that the low-cost monopoly price is the best
collusive price in low-cost periods, the best collusive prices are procyclical. This is
because a small deviation from pm in a low-cost period that is matched in all future
periods only generates a �rst-order decrease in subsequent high-cost periods. This is
(weakly) smaller than the �rst-order decrease in subsequent high-cost periods due to
a small deviation from a price slightly above pm in a high-cost period. Thus, given
the initial deviation pro�t is higher in a low-cost period than one of high costs, a
�rm will not deviate once from a price slightly above pm in a high-cost period if it
will not deviate once from pm in a low-cost period.
4.2 Price rigidity and the expected duration of a high-cost phase
The best collusive prices are rigid when costs are su¢ ciently small, but the critical
level depends upon how long high costs persist into the future. To see this point,
de�ne a high-cost phase as a sequence of high-cost periods that begins in a period
where costs change from low to high costs and ends the period before they change
back. The expected duration of a high-cost phase is �1t=1t�(1� �)t�1 = 1=�, whichimplies the lower the probability that costs will change from high level to the low
level, the longer the expected duration of a high-cost phase. Similarly, de�ne a
low-cost phase with an expected duration of 1=�.
14
Proposition 2 For every n � 2 and � 2 (0; 1) where 1��� � � 0, the critical levelof high costs under which the best collusive prices are rigid, c�, is strictly decreasing
in the expected duration of a high-cost phase whether future levels are independent
of or positively correlated with the current level.
An increase in the expected duration of a high-cost phase has a direct e¤ect
that implies, it is less pro�table to deviate once from a price slightly above the best
rigid price in a high-cost period. This is because a small deviation leads to a larger
�rst-order decrease in future pro�ts than before, since future punishment periods
are more likely to be a (less pro�table) high-cost periods. However, there is also an
indirect e¤ect because the best rigid price increases with the expected duration of
a high-cost phase due to the same reason. This and the standard properties of the
underlying competition game imply that it is more pro�table to deviate once from
a price slightly above the best rigid price in a high-cost period. The direct e¤ect
dominates, so the best collusive prices are rigid for smaller cost �uctuations as the
expected duration of a high-cost phase increases. When there is zero correlation
both the direct and indirect e¤ects are larger than under positive correlation (since
the expected duration of a low-cost phase decreases as the expected duration of a
high-cost phase increases), but the direct e¤ect still dominates.
We have already seen that when a high-cost phase is expected to only last one
period, there is price rigidity as long as the high-cost one-shot Nash price can be
supported in a low-cost period (i.e. c� ! bc as � ! 1 and � ! 0). At the other
extreme, as the expected duration of a high-cost phase tends to in�nity, the best
collusive prices are procyclical regardless of the expected duration of a low-cost phase
(i.e. c� ! 0 as � ! 0 8 0 < � < 1). This is because a low-cost period can be followedby low- and high-cost periods, but a high-cost period will only be followed by other
high-cost periods. Consequently, a small deviation from a price slightly above the
rigid price in a high-cost period leads to a larger decrease in subsequent collusive
pro�t than a small deviation from a rigid price in a low-cost period. Therefore, given
the initial deviation pro�t is also lower in a high-cost period than one of low costs,
procyclical prices are supportable by (3).
4.3 Price rigidity and pro�t over the �uctuations
The preceding analysis showed that the best rigid price is larger for more pronounced
�uctuations. This is because a small deviation from a rigid price in a low-cost
period that is matched in all future periods generates a larger �rst-order decrease in
subsequent high-cost periods as the level of high costs increase. Proposition 3 shows
that this implies that there are also general properties for the best collusive pro�ts
15
as �uctuations become more pronounced.
Proposition 3 For any c 2 (0; c�], the best collusive high-cost (low-cost) per-periodpro�t is strictly decreasing (increasing) in c, and the present discounted value of the
best collusive pro�ts is strictly decreasing in c regardless of whether costs are initially
high or low.
As high costs increase, the best rigid price increases which ensures low-cost per-
period pro�ts are strictly increasing in c. However, only a small proportion of
the higher costs are passed on via a higher price, so high-cost per-period pro�ts
are strictly decreasing in c. When the e¤ect is considered over time, the increase
in expected low-cost pro�ts is outweighed by the expected loss in high-cost pro�ts
regardless of whether such pro�ts are considered from a high-cost or low-cost period.
Thus, when costs �uctuate by a moderate degree over time the collusive price is
higher but collusive pro�ts are lower than when they �uctuate to a lesser extent.
This contrasts with cost �uctuations under Nash reversion where prices and pro�ts
can increase with cost �uctuations (see Rimler, 2005).
5 An Example
We complement the above analysis by assuming that demand is derived from the con-
stant elasticity of substitution version of Spence-Dixit-Stiglitz preferences (Spence,
1976; and Dixit and Stiglitz, 1977). We do this for three reasons. First, we want
to show that price rigidity can occur for reasonably large �uctuations in unit costs.
Second, we want to investigate how the degree of product di¤erentiation a¤ects price
rigidity. To the author�s knowledge, there is no other model of collusive price rigidity
that considers this, since both Athey et al (2004) and Hanazono and Yang (2007)
analyse homogeneous products. Third, we want to investigate the relationship be-
tween price rigidity and the number �rms, and this ultimately depends upon the
degree of product di¤erentiation. We use Spence-Dixit-Stiglitz preferences, because
it falls into the class of our general model and it isolates the competitive e¤ects of
product di¤erentiation since there is no market expansion e¤ect.8
A representative consumer�s utility function is U(x) = n1��
1��
�1n�x
1���i
� 1��1���
+m,
where x is the vector of consumption of the n products, m is expenditure on other
goods, � 2 (0; 1) measures the degree of product di¤erentiation, where products are8Spence-Dixit-Stiglitz preferences is one example of di¤erentiated demand analysed by Kühn
and Rimler (2007) for collusion models under Nash reversion and optimal punishment strategies.It has not been analysed for collusion under price-matching punishments before.
16
less di¤erentiated the closer � is to zero, and � 2 (0; 1) is a parameter. It followsfrom this that the direct demand function for �rm i is:
qi(pi;p�i; �; n) =1
np� 1�
i
"n
�j (pi=pj)1�����
# 1��1���
This implies that at equal prices total demand is independent of both the degree of
product di¤erentiation and the number of �rms, i.e. q(p) = p�1=�. It is straightfor-
ward to show that the monopoly price is pm(ct) = ct=(1� �) and that the one-shotNash price is pN (ct; n; �) = ct=
h1� �=
�1 + 1��
�n�1n
�i. To ensure the monopoly
level is above the one-shot Nash price when costs are low, we assume that the level
of low costs is c 2 (0; c), and normalise the high cost relative to the low cost later.For this example, the best rigid price is:
pyL(c; n; �; �; �; �) =c
1��=�1+(1��) 1��
�n�1n
� + ��(c�c)!�1��+(1��) 1��
�n�1n
� (8)
which applies for c < c � bc.9 The �rst term on the right hand-side of (8) is equivalentto the best collusive price analysed in Lu and Wright (2010) and the second term
captures the e¤ect of cost �uctuations. It equals c when products are homogeneous,
and it is everywhere strictly increasing in the degree of product di¤erentiation, �. It
is above the low-cost one-shot Nash price for all 0 < � � 1, and it is above the low-cost monopoly level when products are not substitutable. It is everywhere strictly
decreasing in the number of �rms, n, but it is always above the low-cost one-shot
Nash price even when there is a large number of �rms in the market.10
The price in (8) de�nes the best collusive price in both cost states when costs
are su¢ ciently small. It follows from Proposition 1 that c� = c1�K 2 (c;bc) where:
K =� ��!(1��) 1��
�n�1n�
(1��)(1� ��!)+(1��) 1��
�n�1n
��1� ��
!+(1��) 1��
�n�1n
� 2 (0; 1).
To illustrate the properties of c�, Figure 2 plots �c� � c��cc = K
1�K as a function
of � for three levels of n. This has two intrepretations. First, �c� is the critical
proportion that high costs can be above low costs under which the best collusive
prices are rigid. Second, it measures the proportion to which prices would vary if
the market were monopolised (i.e. [pm(c�)�pm(c)]=pm(c) = �c�). Parameter valuesare chosen such that the low-cost monopoly level is equal to unity, and that future
costs are independent of the current level and are equally likely.
The Figure shows that there is a non-monotonic relationship between �c� and
9Following Lemma 1, bc = �1� �= h1 + 1���
n�1n
i�c
1��(1� ��!)=h(1��)
�1+ 1��
�n�1n
�+�(1� �
!)i .
10The reason is that the underlying competition game is one of monopolistic competition.
17
Figure 2: (c = 0:5; � = 1� � = 0:5, � = 0:5, � = 0:9)
the degree of product di¤erentiation. For intermediate degrees of di¤erentiation, the
best collusive prices are rigid when the high-cost monopoly level is 16% higher than
the low-cost monopoly price, and price rigidity can occur for even larger �uctuations
when the expected duration of a high-cost phase is shorter.11 When products are
close to perfect substitutes or when they are barely substitutable, the best collusive
prices are rigid for small �uctuations. This is because the punishment strategy does
not support collusive prices if the products are homogeneous, since an in�ntesimally
small deviation from the collusive price captures the whole market and the price-
matching strategy provides virtually no punishment. Consequently, �rms price at
the one-shot Nash level in each cost state and prices are procyclical. In contrast,
when �rms are local monopolies there is not price rigidity, because each monopoly
is unconcerned with the reactions of the other �rms, so they price at the relevant
monopoly levels in each cost state.
The Figure also shows that �c� decreases with the number of �rms for low
levels of di¤erentiation, but the opposite relationship can exist otherwise. This
non-monotonic relationship between price rigidity and the number of �rms is con-
sistent with Hanazono and Yang (2007), and it is not inconsistent with empirical
research that shows the responsiveness of prices to �uctuations is negatively corre-
lated with concentration in some cases (see Dixon, 1983; Carlton, 1986; Bedrossian
and Moschos, 1988; Geroski, 1992; Weiss, 1995) and the opposite relationship exists
11For example, if � = 0:01 and � = 0:99, the shape of �c� is similar to that of Figure 2 exceptthat the best collusive prices are rigid when the high-cost monopoly price is 40% higher than thelow-cost monopoly price.
18
in others (see Domberger, 1979; and Kardasz and Stollery, 1988). In our general
framework, the relationship is unsignable, because a direct e¤ect increases a �rm�s
incentive to deviate once from a price slightly above the best rigid price in a high-
cost period, but an indirect e¤ect works in the opposite direction because the best
rigid price is strictly decreasing in n (due to Assumptions 4 and 5). When more
structure is placed on demand, the indirect e¤ect at the homogeneous goods limit
is equal to the change in a �rm�s incentive to deviate once from the best rigid price
in a low-cost period. Given an in�nitesimally small deviation attracts the whole
market, the changes in punishment pro�ts are arbitrarily small. Thus, the indirect
e¤ect dominates, because due to lower costs an extra �rm has a larger e¤ect on the
initial deviation pro�t in a low-cost period than that in a high-cost period. As a
result, procyclical prices are easier to support at the homogeneous good limit as the
number of �rms increases.
6 Concluding remarks
In this paper we have analysed an in�nitely repeated game where unit costs �uctuate
stochastically over time and �rms employ a price-matching punishment strategy.
We showed that the best collusive prices can be rigid over time when the cost
�uctuations are su¢ ciently small. This provides game theoretic support for the
results of the kinked demand curve. Speci�cally, when a period of high costs is
expected to be followed by low-cost periods only, a price above the low-cost price
cannot be supported when the high-cost one-shot Nash equilibrium is below the low-
cost price. The critical level of high costs under which the best collusive prices are
rigid is monitonically decreasing in the expected duration of a high-cost phase, and
price rigidity does not occur when costs are expected to increase permanently in the
future. Each �rm�s present discounted value of the best collusive pro�ts decreases as
the cost �uctuations become more pronounced, despite an increase in the best rigid
price. When demand is derived from the constant elasticity of substitution version
of Spence-Dixit-Stiglitz preferences, the best collusive prices are rigid for reasonably
large cost �uctuations when products are di¤erentiated by an intermediate degree;
and when products are su¢ ciently substitutable, price rigidity occurs for larger cost
�uctuations as the number of �rms in the market decreases.
Throughout the paper we have considered only two cost states, but periods of
price rigidity are not restricted to this special case. For instance, when a medium
cost state is added and the probabilities of future costs are independent of the
current level, the best rigid price is una¤ected by the introduction of the third state
if the expected level of future costs is unchanged compared to the two-state model.
This also implies that the punishment from high-cost periods is lower than before,
19
because from holding the expected level of future costs constant, future high-cost
periods are less likely in the three-state model than the two-state. As a result, it is
more di¢ cult to support a procyclical price in high-cost periods than before. Thus,
for �uctuations where best collusive prices are rigid in the two-state model, the best
collusive prices in the three-state model will either be rigid for every cost state or
partially rigid (where there is price rigidity in medium and high costs states but
at a price above that set in a low-cost period). Applying this logic to more than
three cost states implies that it is even more di¢ cult to support procyclical prices
in the highest-cost state, so periods of price rigidity can also occur for any number
of states.
Finally, an important avenue for future research is to investigate whether there
exist any circumstances under which �rms will choose to support collusive prices
through a weaker punishment, such as price matching, rather than revert to harsher
punishment strategies, such as Nash reversion or optimal punishment strategies.
As well as resolving the tension between the informal reasoning that �rms would
prefer to follow the harshest punishment possible with the evidence that, at least in
some situations, tacitly colluding �rms (and even some cartels) do not follow harsh
punishments, a theoretical justi�cation for price matching may provide a better
indication of the industry characteristics where price rigidity is likely to prevail.
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A Proofs
Proof of Lemma 1. To �nd the conditions under which setting pN (c; n) in both
cost states is supportable by (3), it su¢ ces to check that a �rm will not deviate
once in the collusive subgame and nor will it do so in every possible punishment
subgame. For every history that starts with �rms setting pN (c; n) in both cost
states, the relevant deviation price at some period � is min(pd� ; pN (c; n)). In high-
cost subgames, (3) trivally de�nes an equilibrium. Likewise, in low-cost subgames,
(3) de�nes an equilibrium if min(pd� ) � pN (n). Otherwise, all other histories lead to
22
subgames equivalent to one where �rms should set pN (c; n) and min(pd� ; pN (c; n)) 2
(pN (n); pN (c; n)] in high- and low-cost periods, respectively.
Since (3) de�nes an equilibrium in every high-cost subgame, suppose we consider
a low-cost subgame where prices are p 2 (pN (n); pN (c; n)] and pN (c; n) in low- andhigh-cost periods, respectively. From (4) de�ne:
�zL(p) �h@�i(pi;p;n)
@pi+ �
1�� (1��! )
d�(pi;n)dpi
ipi=p
.
A �rm will not deviate once from pN (c; n) in the low-cost collusive subgame if
�zL(pN (c; n)) � 0, otherwise it can increase its pro�t by unilaterally lowering its
price from pN (c; n). We wish to show that �zL(p) � 0 8 p 2 (pN (n); pN (c; n)).Di¤erentiating �zL(p) with respect to p yields:
d(�zL(p))dp =
h@2�i(pi;p;n)
@p2i+Pj 6=i
@2�i(pi;p;n)@pi@pj
+ �1�� (1�
�! )
d2�(pi;n)dp2i
ipi=pj=p
.
It follows from (2) and Assumption 5 that d(�zL(p))=dp < 0. Hence, if�zL(p
N (c; n)) �0, then �zL(p) > 0 8 p 2 (pN (n); pN (c; n)). Thus, a �rm will not deviate once froma price between pN (n) and pN (c; n) in any low-cost punishment subgame.
There exists a unique bc 2 (0; c) such that�zL(pN (bc; n)) = 0 because�zL(pN (n)) >0, �zL(p
N (c; n)) = �zL(pm) < 0 and:
d(�zL(pN ))
dc =hd(�zL(p))
dpdpN
dc
ip=pN (c;n)
< 0,
since d(�zL(p))=dp < 0 and dpN=dc > 0. Therefore, �zL(p
N (c; n)) � 0 if and onlyif c 2 (0;bc]. The above analysis implies, any prices where �rms set pN (c; n) andp, such that pN (n) < p � pN (c; n), in high- and low-cost periods, respectively, aresupportable by (3) if and only if c 2 (0;bc].Proof of Lemma 2. To �nd the conditions under which setting pc in both
cost states is supportable by (3), it su¢ ces to check that a �rm will not deviate
once in the collusive subgame and nor will it do so in every possible punishment
subgame. For every history that starts with �rms setting pc in both cost states, the
relevant deviation price at some period � is min(pd� ; pc). If min(pd� ; p
c) � pN (c; n),(3) de�nes an equilibrium in such subgames if and only if c 2 (0;bc] (see Lemma 1).Otherwise, all other histories lead to subgames equivalent to one where �rms should
set min(pd� ; pc) 2 (pN (c; n); pc] in high- and low-cost periods.
Suppose we consider a subgame where c 2 (0;bc] and the price is p 2 (pN (c; n); pc]in both cost states. When costs are low, it is more pro�table to deviate once from p
to pN (c; n) than to any price below pN (c; n), because �zL(pN (c; n)) � 0 and prices
are strategic complements. Thus, consider �rm i�s incentives to set pi 2 [pN (c; n); p]
23
in low- and high-cost subgames. From (5) and (6), respectively, de�ne:
�yH(p) �h@�i(pi;p;c;n)
@pi+ �
1��
��!d�(pi;n)dpi
+ [1� �! ]d�(pi;c;n)
dpi
�ipi=p
�yL(p) �h@�i(pi;p;n)
@pi+ �
1��
��!d�(pi;c;n)
dpi+ [1� �
! ]d�(pi;n)dpi
�ipi=p
.
A �rm will not deviate once from pc in the low-cost collusive subgame if�yL(pc) � 0,
otherwise it can increase its pro�t by unilaterally lowering its price from pc. We wish
to show that �yL(p) � 0 and �yH(p) � 0 8 p 2 (pN (c; n); pc]. First, di¤erentiating
�yL(p) with respect to p yields:
d(�yL(p))dp =
h@2�i(pi;p;n)
@p2i+Pj 6=i
@2�i(pi;p;n)@pi@pj
+ �1��
��!d2�(pi;c;n)
dp2i+ [1� �
! ]d2�(pi;n)dp2i
�ipi=pj=p
.
It follows from (2) and Assumption 5 that d(�yL(p))=dp < 0. Hence, if �yL(p
c) �0, then �yL(p) > 0 8 p 2 (pN (c; n); pc). Next, consider:
�yH(p)��yL(p) = �
hc@qi(pi;p;n)@pi
+ �! (1� � � �)
cndq(pi)dpi
ipi=p
.
Assumption 1 and 1���� � 0 imply that the above is positive. So, if �yL(pc) � 0,then �yH(p) > �
yL(p) � 0 8 p 2 (pN (c; n); pc]. Thus, a �rm also will not deviate
once from pc in the high-cost collusive subgame, and nor will it do so from a price
between pN (c; n) and pc in any low- or high-cost punishment subgame.
Given d(�yL(p))=dp < 0, there exists a unique best rigid price, pyL(c; n; �; �; �),
that solves �yL(p) = 0. It satis�es pN (c; n) < pyL(c; n; �; �; �) < pm(c) since
�yL(pN (c; n)) > �zL(p
N (c; n)) � 0 8 c 2 (0;bc] and �yL(pm(c)) < 0. The aboveanalysis implies, any rigid price pc such that pN (c; n) < pc � pyL(c; n; �; �; �) is
supportable by (3) if and only if c 2 (0;bc].Proof of Proposition 1. To �nd the conditions under which setting p(0) and
p(c) > p(0) in low- and high-cost periods, respectively, is supportable by (3), it suf-
�ces to check that a �rm will not deviate once in the collusive subgame and nor will it
do so in every possible punishment subgame. For every history that starts with �rms
setting p(0) and p(c) > p(0) in low- and high-cost periods, the relevant deviation
price at some period � is min(pd� ; p(c� )). If min(pd� ) � p(0), then for any c 2 (0;bc],
(3) de�nes an equilibrium in such subgames if and only if p(0) � pyL(c; n; �; �; �) (seeLemma 2). Otherwise, all other histories lead to subgames equivalent to one where
�rms should set max(pN (c; n);min(pd� ; p(c))) 2 (max(pN (c; n); p(0)); p(c)] and p(0)in high- and low-cost periods, respectively.
Suppose we consider a low-cost subgame where c 2 (0;bc] and the prices arep 2 (p(0); p(c)] and p(0) in high- and low-cost periods, respectively, where without
24
loss of generality let p(0) � pN (c; n). A �rm cannot increase its pro�t by deviating
once to a price above p(0), and deviating once to a price below p(0) is less pro�table
when prices are procyclical than when they are rigid. Thus, any p(0) such that
pN (c; n) < p(0) � pyL(c; n; �; �; �) is supportable by (3), and the best collusive pricein low-cost periods is p�(0) = min(pyL(c; n; �; �; �); p
m) 8 c 2 (0;bc].Now consider a high-cost subgame where c 2 (0;bc] and the prices are p 2
(p�(0); p(c)] and p�(0) in high- and low-cost periods, respectively. It is more prof-
itable to deviate once from p to p�(0) than to a price below p�(0) in a high-cost
period, because �yH(p�(0)) > �yL(p
�(0)) � 0 and prices are strategic comple-
ments. Thus, consider �rm i�s incentive to set pi 2 [p�(0); p]. From (7), de�ne:
�xH(p) �h@�i(pi;p;c;n)
@pi+ �
1�� (1��! )
d�(pi;c;n)dpi
ipi=p
.
A �rm will not deviate once from p(c) in the high-cost subgame if �xH(p(c)) � 0,otherwise it increases its pro�t by unilaterally lowering its price from p(c). We wish
to �nd conditions under which �xH(p) � 0 8 p 2 (p�(0); p(c)]. Di¤erentiating
�xH(p) with respect to p yields:
d(�xH(p))dp =
h@2�i(pi;p;c;n)
@p2i+Pj 6=i
@2�i(pi;p;c;n)@pi@pj
+ �1�� (1�
�! )
d2�(pi;c;n)dp2i
ipi=pj=p
.
It follows from (2) and Assumption 5 that d(�xH(p))=dp < 0. Hence, if�xH(p(c)) �
0, then �xH(p) > 0 8 p 2 (p�(0); p(c)]. Thus, a �rm also will not deviate once from
a price between p�(0) and p(c) in any high-cost punishment subgame. This implies,
the best rigid price is also the best collusive price in a high-cost period if and only
if �xH(pyL(c; n; �; �; �)) � 0, where p
yL(c; n; �; �; �) < p
m since �xH(pm) > 0 when
�yL(pm) � 0.
To prove there exists a unique c� 2 (0;bc) such that �xH(pyL(c�; n; �; �; �)) = 0,�rst notice that c� > 0 since limc!0�xH(p
c) < limc!0�yL(p
c) 8 pc 2 [pN (c; n); pm].Next, consider:
d(�xH(pyL))
dc = �h@qi(pi;p;n)
@pi+ �
(1��)1n [1�
�! ]dq(pi)dpi
� d(�xH(p))dp
dpyLdc
ipi=p=p
yL(c;n;�;�;�)
= �h@qi(pi;p;n)
@pi+ �
(1��)1n [1�
�! � �
�! ]dq(pi)dpi
ipi=p=p
yL(c;n;�;�;�)
where � � [d(�xH(p))=dp]=[d(�yL(p))=dp] > 0 and:
dpyLdc =
1d(�
yL(p))
dp
�(1��)
1n�!dq(pi)dpi
���pi=p=p
yL(c;n;�;�;�)
> 0.
If [1� �! � �
�! ] > 0, then d(�
xH(p
yL))=dc > 0 and � � 1 is su¢ cient for this to be
25
true. Subtracting d(�yL(p))=dp from d(�xH(p))=dp yields:
�hc�@2qi(pi;p;n)
@p2i+Pj 6=i
@2qi(pi;p;n)@pi@pj
�+ �
1�� (1��! )
d2�(pi;n)dp2i
+ �! (1� � � �)
d2�(pi;c;n)dp2i
ipi=pj=p
,
(9)
which is positive from Assumptions 3 and 5, and 1 � � � � � 0, so � < 1. Hence,d(�xH(p
yL))=dc > 0, which implies c� is unique and �xH(p
yL(c; n; �; �; �)) < 0 8
c 2 (0; c�). Finally, to see that c� < bc, consider:�xH(p)��
yL(p) =
h@�i(pi;p;c;n)
@pi+ �
! (1� � � �)d�(pi;c;n)
dpi��zL(p)
ipi=p
. (10)
When evaluated at pyL(c; n; �; �; �), (10) is non-positive 8 c 2 (0; c�]. Di¤erentiating(10) with respect to p yields (9), which is positive. This implies that (10) is negative
8 p 2 [pN (c; n); pyL(c; n; �; �; �)). Thus, for (10) to be negative when evaluated atpN (c; n), it follows that �zL(p
N (c; n)) > 0 since the �rst term on the right-hand
side of (10) is zero and the second is non-negative when 1 � � � � � 0. Given
�zL(pN (bc; n)) = 0, then c� < bc since d(�zL(pN ))=dc < 0 (see Lemma 1).
Proof of Proposition 2. Totalling di¤erentiating �xH(pyL(c; n; �; �; �)) = 0
yields:dc�
d� = � 1d(�x
H(pyL))
dc
d(�xH(pyL))
d�
dc�
d� = � 1d(�x
H(pyL))
dc
d(�xH(pyL))
d�
where d(�xH(pyL))=dc > 0 from Proposition 1.
The total derivative of �xH(pyL(c; n; �; �; �)) with respect to � when � = 1 � �
is:d(�xH(p
yL))
d� =h�1��
d�(pi;c;n)dpi
+d(�xH(p))
dpdpyLd�
ipi=p
yL(c;n;�;�;�)
= �1��
hd�(pi;n)dpi
� (1� �) cndq(pi)dpi
ipi=p
yL(c;n;�;�;�)
where:dpyLd� =
1d(�
yL(p))
dp
�(1��)
cndq(pi)dpi
���pi=p=p
yL(c;n;�;�)
> 0.
The total derivative of �xH(pyL(c; n; �; �; �)) with respect to � is:
d(�xH(pyL))
d� = �h�(1��(1��))(1��)!2
d�(pi;c;n)dpi
� d(�xH(p))dp
dpyLd�
ipi=p
yL(c;n;�;�;�)
= � �(1��)!2
h(1� �[1� �(1� �)])d�(pi;c;n)dpi
+ ���d�(pi;n)dpi
ipi=p
yL(c;n;�;�;�)
where:dpyLd� = �
1d(�
yL(p))
dp
�2�(1��)!2
cndq(pi)dpi
���pi=p=p
yL(c;n;�;�;�)
< 0.
Thus, it follows from Assumptions 1 and 5, and 0 < � < 1 (see Proposition 1) that
26
d(�xH(pyL))=d� > 0 and d(�
xH(p
yL))=d� < 0, so dc
�=d� < 0 when � = 1 � � anddc�=d� > 0.
Proof of Proposition 3. Assumptions 1, 3 and 5 imply that dpyL=dc 2 (0; �=!).This guarantees that when per-period pro�ts are evaluated at pyL(c; n; �; �; �), they
are strictly increasing in c when costs are low but they are strictly decreasing in c
when costs are high.
Next, evaluate p and pc in (6) at pyL(c; n; �; �; �) and totally di¤erentiate with
respect to c. After some rearranging, this yields:
� �1��
�!
"q(p)n + (p�c)
ndq(p)dp
1
�d(�
yL(p))
dp
�j 6=i@q(p;p;n)@pj
#p=pyL(c;n;�;�;�)
.
The above is negative since d�(p;c;n)dp = q(p)
n + (p�c)n
dq(p)dp > 0 when p is evaluated
at pyL(c; n; �; �; �) 2 (pN (c; n); pm), and���d(�yL(p))dp
��� > ���dq(pi)dpi
��� > �j 6=i@q(p;p;n)@pj
. This
implies, each �rm�s present discounted value of collusive pro�ts in a low-cost period
is decreasing in c.
Finally, evaluate p and pc in (5) at pyL(c; n; �; �; �) and totally di¤erentiate with
respect to c. After some rearranging, this yields:
�hq(p)n
�1� dpyL
dc
�+ q(p)
n�1��
�1� �=! � dpyL
dc
�� dq(p)
dp
�(p� c) + �
1�� (p� (1��! )c)
�dpyLdc
ip=pyL(c;n;�;�;�)
.
It follows from Assumption 1 and from 0 < dpyL=dc < �=! � (1 � �=!) < 1 when
1���� � 0 that the above is negative. This implies, each �rm�s present discountedvalue of collusive pro�ts in a high-cost period is decreasing in c.
27