Collusion with intertemporal price dispersion*
Nicolas de Roos†
Sydney University
Vladimir Smirnov‡
Sydney University
Abstract
We develop a theory of optimal collusive intertemporal price dispersion. The extent of
dispersion influences consumer awareness of individual prices, providing an incentive
for firms to coordinate on dispersed prices. Our theory generates a collusive rationale
for price cycles and sales. As in the perfect information setting, optimal collusion at
the monopoly price can be supported if firms are sufficiently patient. For lower discount factors, monopoly prices must be punctuated with fleeting sales. The most robust
structure involves asymmetric price cycles that resemble Edgeworth cycles. If consumer
attentiveness is low, the effectiveness of price dispersion is enhanced by reducing the
payoff to deviations involving price reductions. However, for sufficiently low attentiveness, price rises are also a concern, limiting the power of obfuscation.
JEL Classification: L13, D83
Keywords: Collusion, obfuscation, price dispersion.
The nagging presence of competitors is an inescapable fact of life for most firms. To
understand pricing behaviour, we must therefore consider the implications of repeated interaction. The theory of repeated games offers some insights; notably, repeated play opens
up opportunities for punishments and rewards, permitting higher prices to be supported.
However, non-trivial price dynamics are also a common outcome of repeated play. For example, sales are pervasive and, in retail petrol markets, regular price cycles are a common
feature. The rationale for these pricing patterns is less obvious. If the threat of punishment
is available, why don’t firms employ it to support the highest feasible fixed price? Complicated pricing patterns may be both more difficult to coordinate on and less profitable. In
this paper we develop a theory of coordinated price dispersion that provides an intuitive
* We
are grateful to Murali Agastya, Garry Barrett, Michelle Bergemann, Mark Melatos, Debraj Rey, John Romalis, Abhijit Sengupta, Kunal Sengupta, Jeroen van de Ven, and Andrew Wait for useful comments, suggestions
and encouragement. We thank seminar audiences at the Australian National University, the University of New
South Wales, the University of Sydney, and the University of Queensland.
†
Corresponding author: School of Economics, Merewether Building H04, University of Sydney, NSW 2006,
Australia, e-mail: nicolas.deroos@sydney.edu.au, phone: +61 (2) 9351 7079, fax: +61 (2) 9351 4341.
‡
School of Economics, Merewether Building H04, University of Sydney, NSW 2006, Australia, e-mail:
vladimir.smirnov@sydney.edu.au.
1
explanation. Our theory rationalises commonly observed pricing patterns including sales,
price cycles, and fixed prices. Our starting point is the consumer.
Pretend for a moment that you are a consumer. Think of a few products that may be in
your shopping basket; for example, milk, coffee, petrol, breakfast cereals. What is the current price of these items at your local store? What is the current price at other stores? Would
you recognise a bargain? Your answers to these questions may vary by product; frequency of
purchase and prominence of display are obvious factors. We conjecture that the complexity
of pricing patterns also plays a role. A consumer who routinely observes a single price may
become accustomed to that price and recognise immediately a departure from this simple
pricing pattern. By contrast, complicated price paths are more difficult to absorb, making
price changes less obvious.1 If consumer price perceptions are influenced by pricing patterns in this manner, they may be ripe for manipulation by firms. We consider the problem
of a cartel faced with this prospect.
In our model, firms engage in repeated simultaneous price competition for a homogeneous product. We seek the optimal collusive pricing strategy. The immediate problem for
any cartel is to maintain incentives for internal discipline. Each cartel member trades off
the short-term incentive to deviate and raise current profits against the long-term benefit
of receiving lucrative cartel payoffs. We introduce imperfectly attentive consumers to this
standard environment. The propensity for a consumer to observe and recall a specific price
depends on the prices she experiences. If prices are intertemporally dispersed and she is
accustomed to seeing prices over a wide interval, then she may not be responsive to price
differences within this interval. By contrast, if prices are fixed over time, even modest price
changes might trigger her attention. Price dispersion then plays a useful role for the cartel
as an obfuscation device. With greater dispersion, consumers are less responsive to price
changes, the payoff to deviating from cartel policies is reduced, and the cartel’s internal incentive constraints are relaxed. The optimal dynamic price path for the cartel then emerges
as a trade off between profitability and obfuscation.
In addition to cartel manipulation, innate market characteristics influence a consumer’s
tendency to perceive and recall price information. We parameterise our model by the level of
consumer attentiveness, ranging from perfect awareness to complete inattentiveness. At one
extreme, our specification approaches the Bertrand model and obfuscation is futile. Consumers are aware of all prices and an undercutting firm captures the whole market independent of the extent of intertemporal price dispersion. At the other extreme, if consumers are
not aware of specific prices, our model can replicate the Diamond (1971) paradox: if all firms
were to set a price below the monopoly price in a single-period version of our model, then
there would be an incentive to raise price.
Because we study infinitely repeated play, our setting admits a spectrum of equilibria.
We examine optimal symmetric collusive equilibria with strategies of finite complexity. The
price paths we consider are infinitely repeated sequences of finite length. Restriction to finite
1
This line of reasoning is supported by the concepts of rehearsal and associativeness that Mullainathan
(2002) draws from the literature on memory research. Taken together, these two principles may imply a constant price would become a well established reference for comparison, while an intertemporally dispersed
price path might be less amenable to these memory processes.
2
strategies has an intuitive rationale. Costs of negotiation, coordination and enforcement are
likely to rise for a cartel with the complexity of the price path, placing a limit on the optimal
complexity from the cartel’s perspective.
Let us briefly preview some of our main findings. Our most fundamental result is that
collusive equilibria exist for a greater range of discount factors with an intertemporally dispersed price path than with a fixed price. The optimal price path reflects a compromise between profitability and the obfuscation properties of the path. Every sequence begins with
the monopoly price and then follows a weakly decreasing trajectory; each subsequent period involves either monopoly pricing or a strict price reduction. If cartel members are sufficiently patient, setting a fixed price at the monopoly level is optimal. However, for lower
values of the discount factor, intertemporally dispersed prices are required to dampen the
market share benefits for potential deviators. Two specific equilibrium outcomes of our
model are closely related to commonly observed pricing dynamics. First, for a range of parameter values, we observe sales: prolonged periods of monopoly pricing interspersed with
large temporary price reductions. One-period sales present the cheapest means of delivering
price dispersion. If firms are less patient, the length and depth of sales must increase in order
to satisfy the incentive constraints of the cartel. Second, sales of maximum length and depth
afford the greatest protection against deviation. These pricing paths resemble asymmetric
price cycles known as Edgeworth cycles.
In markets with alert consumers the benefits of price dispersion for cartel sustainability are minimal. As attentiveness is reduced, obfuscation becomes more effective and cartel
sustainability is improved. However, there is a limit to this process. For low levels of attentiveness, the primary threat to cartel stability is not the temptation to undercut, but rather
the incentive to raise prices above the levels dictated by the cartel. If consumers pay little
heed to prices, then the penalty for doing so may be minimal. Consequently, if attentiveness
levels are sufficiently low, the relationship between cartel sustainability and attentiveness
may be reversed: as attentiveness is reduced, deviation to monopoly pricing becomes more
attractive and the cartel becomes harder to sustain.
The idea that consumers are imperfectly informed about prices is not new. For example, a substantial literature on consumer search demonstrates that imperfect information
can have dramatic consequences for pricing.2 If consumers are imperfectly informed about
prices, this disturbs a fundamental pricing trade-off faced by firms: prices balance the lure
of extracting rents from consumers with the incentive to wrest consumers from competitors
and defend against rival behaviour. If consumers are poorly informed, the balance may be
tilted towards rent extraction and higher prices. Our departure is to allow a consumer’s price
awareness to be coloured by her experience with prices. Unusual or attractive prices, relative to experience, are more likely to trigger consumer attention. Firms can then collectively
2
See Baye et al. (2006) for a recent survey. Note that the interplay between collusion and price dispersion in
our model is quite different to a typical consumer search setting. A common assumption in consumer search is
that consumers are perfectly informed about the current price distribution. Price dispersion is then potentially
harmful for a cartel because it raises the benefits of search, improving consumer price awareness and making
the cartel’s incentive constraints more difficult to satisfy. By contrast, in our theory, price dispersion reduces
consumer price awareness, relaxing the cartel’s incentive constraints.
3
adjust the pricing trade-off by influencing the history of prices consumers experience.
Our specification of consumer behaviour is related to the theory of rational inattention
(Sims, 2003) and Range Theory (Volkmann, 1951). Rationally inattentive consumers have
limited information processing capacity, and optimally deploy this capacity.3 If more complicated price paths require greater information processing, then this may reduce the effectiveness of price comparison. Range Theory suggests that the attractiveness of a specific
market price is relative to the lower and upper bound of price expectations consumers form
through experience.4 Similarly, consumers in our model are more likely to retain the price
information of a specific vendor if the offered price is unusual relative to their experience.
Our work contributes to a growing literature on obfuscation in oligopoly. Obfuscation operates by hampering consumer efforts to directly compare competing products. For example, obfuscation could take the form of additional noise in the price process (Spiegler, 2006),
the choice of price formats with different comparability properties (Piccione and Spiegler,
2012; Chioveanu and Zhou, 2013), or attempts to raise the cost of consumer search (Ellison
and Wolitzky, 2012; Wilson, 2010).5 Our principal conceptual departure from this literature
is to offer a motivation for coordinated obfuscation. Even if cartel members are able to coordinate on prices, we illustrate that coordination on obfuscation may be required to relax
the incentive constraints of the cartel. In our theory, intertemporal price variation acts as the
obfuscation mechanism; we expect alternative mechanisms to yield similar insights.
We are not the first to examine cartel pricing dynamics. However, most explanations
for cartel pricing dynamics involve exogenous market processes; examples include demand
side dynamics (Green and Porter, 1984; Rotemberg and Saloner, 1986), and entry, exit, and
investment dynamics (Fershtman and Pakes, 2000; de Roos, 2004).6 We demonstrate that,
even when faced with identical consumers and a stationary market environment, a cartel’s
optimal price path may involve non-trivial price dynamics.7 Our characterisation of the op3
Recent examples suggest inattention can be extreme. Clerides and Courty (2010) document that some
consumers do not make basic price comparisons even when the information is right before them. Monroe and
Lee (1999) argue that consumers often do not remember specific prices for products they have purchased, but
they nevertheless have an intuitive understanding about prices.
4
Range Theory has some recent observational support. For example, Janiszewski and Lichtenstein (1999)
present experimental evidence that variation in the width of the price range affects price-attractiveness judgments, independent of changes to a consumer’s reference price. Relatedly, Moon and Voss (2009) find that a
model based on range theory provides additional explanatory power in the presence of reference price theories
in a panel data setting.
5
In Appendix B, we introduce a consumer search model as a possible foundation of our model. Consumers
sample from the available offerings and recall attention-grabbing prices for future consideration. This mechanism is most closely related to the setting considered by Spiegler (2006), in which consumers adopt a simple
search heuristic: they sample the price of each firm, identify the cheapest offering, and then return to buy from
this firm. Unfortunately (for consumers), firms adopt dispersed prices in equilibrium, and the firm consumers
return to may no longer offer the cheapest price.
6
The only other example we are aware of is the cartel’s optimal pricing problem in the presence of an antitrust authority (Harrington, 2004, 2005). In that case, price dynamics are temporary as the cartel transitions
towards optimal collusive pricing under the watchful eyes of the anti-trust authority. By contrast, we observe
lasting intertemporal price variation.
7
In our theory, firms are intertemporal optimisers and consumers behave myopically. Fershtman and Fishman (1992) demonstrate that non-trivial price dynamics can arise when these roles are reversed.
4
timal path leads to two widely observed structures as special cases: sales and Edgeworth
cycles.
Sales are often associated with price discrimination.8 For example, Sobel (1984) suggests
sales as a method to discriminate between customers with high and low reservation prices,
while in Varian (1980), sales discriminate between informed and uninformed consumers. We
suggest obfuscation as an alternative motivation.9 A brief sale can generate a wide interval
of prices, hampering consumer efforts at price comparison at minimal cost to the cartel.
We also provide an explanation for price cycles. Asymmetric cycles that resemble Edgeworth cycles provide the greatest internal stability for the cartel. Edgeworth cycles are a
striking feature of retail petrol markets in a number of countries including Canada (Noel,
2007; Eckert and West, 2004), the United States (Lewis, forthcoming), Australia (Wang, 2009;
de Roos and Katayama, 2013), and Norway (Foros and Steen, 2008). The most commonly
cited explanation for the Edgeworth cycle is the price commitment model of Maskin and Tirole (1988).10 Asynchronous price setting and the restriction to Markov strategies are central
to the predictions of their model. These features are absent in our model. Instead, coordination and simultaneous play are key ingredients.
There is growing empirical support for coordination and simultaneous play in retail petrol
markets. First, the limited evidence on the timing of play in markets with retail price cycles suggests simultaneous price setting (de Roos and Katayama, 2013). Second, the variety of coordinating mechanisms used suggests coordination plays a central role. A focal
day of the week has been employed in Australian cities (Australian Competition and Consumer Commission, 2007) and Norway (Foros and Steen, 2008). Price leadership by a single
brand features in Canada (Byrne et al., 2013), the United States (Lewis, forthcoming), and
Australia (de Roos and Katayama, 2013). Mixed strategies briefly played a role in Perth, Australia (Wang, 2009). Temporary price spikes have been used to signal the start of a new cycle
in Ontario (Atkinson, 2009). Finally, explicit communication is evident in Ballarat, Australia
(Wang, 2008) and Québec (Clark and Houde, 2011).
The rest of the paper is structured as follows. The main body of the paper is contained
in Section 1, where we introduce the model, and solve the cartel’s optimal dynamic pricing
problem. The character of the solution depends on the magnitude of a salience parameter
indexing the level of consumer attentiveness. We break our results into three subsections
considering high- and low-salience environments, respectively, before examining the optimal complexity of the price path for the cartel. In Section 2, we briefly discuss the main
assumptions of the model and consider some extensions which we leave for future research.
8
Other explanations include demand anticipation (Salop and Stiglitz, 1982) and introductory offers for
experience goods (Doyle, 1986).
9
While there is no role for price discrimination in our model, obfuscation and price discrimination motives
for sales could be complementary.
10
A class of alternative explanations for the cycle includes the original Edgeworth (1925) model of capacity
constrained price competition. The principle components are a discontinuity in residual demand and positive
residual demand above the discontinuity. A potential limitation of this class of explanation is that, to explain a
cycle as an equilibrium phenomenon, we must assume that firms play a myopic best response to their rivals’
previous prices. See de Roos (2012) for details.
5
We conclude with Section 3. In the Appendices, we detail all proofs, introduce a consumer
search model consistent with our theory, and consider the implications of more general
profit functions.
1 The model
A market comprises a set of n firms, indexed by j = 1, . . . , n, selling an undifferentiated product to a unit mass of consumers in each time period t =R 0, . . . , ∞. Each consumer has a
∞
downward-sloping demand function given by D(p) with 0 D(x) d x < ∞, and associated
differentiable revenue function R(p) = pD(p). Consumers are not strategic players; they myopically maximise surplus in the current period. Firms discount the future at the common
rate δ ∈ (0, 1) and have constant marginal costs which we normalise to zero. Assumption 1
places mild restrictions on the demand function. For future reference, we will refer to the
price p̂ as the monopoly price.
Assumption 1. The revenue function R(p) is differentiable, attains a unique maximum at
price p̂, and satisfies R 0 (p) > 0 for p < p̂.
Consumers are imperfectly informed about prices. In particular, consumers have an impression of the price distribution, but they do not observe and recall individual prices unless
they are unusual or attractive. This is summarised by the following market share function,
where we define market shares in terms of the share of consumers rather than output. Suppose consumer beliefs about the lower and upper bounds of the support of the price distribution are given by p and p, respectively, and suppose that firm j sets price p j while her
rivals all set price p s . Let p ≡ (p 1 , p 2 , . . . , p n ) denote a price vector. The market share for firm
j is then given by


0,
if p s < min{p, p j } or p j > max{p s , p}






γ(β), if p ≤ p s < p j ≤ p
s j (p, p, p) = n1 ,
(1)
if p j = p s




α(β), if p ≤ p j < p s ≤ p




1,
if p j < min{p, p s } or p s > max{p j , p}
where γ(β) ∈ [0, 1/n] and α(β) ∈ [1/n, 1] describe market shares for high-priced and lowpriced firms, respectively. Intuitively, if firm j sets a price higher than her rivals, but within
the realm of experience of consumers, then not all consumers will detect and recall this price
difference. Hence, she may still attract a positive measure of consumers. Alternatively, if she
sets a lower price than her rivals and the price difference is modest, she may not attract
the attention of all consumers. If she sets a price outside the bounds p and p, consumers
are startled to attention and either flock to her store (if p j < p) or studiously avoid her (if
p j > p).11
11
In our Example 1 below, we provide an illustrative model to generate such a market share relation. In
Appendix B, we provide a more formal model of consumer search that gives rise to (1).
6
The ability of consumers to observe and recall market prices may be determined by
market-specific characteristics. With a view to comparative statics, we let β index consumer
powers of observation and recall and allow this parameter to determine the market shares
γ(β) and α(β). In markets in which consumers are better able to distinguish prices, there is
a greater market share payoff to undercutting the prices of one’s rivals, and a smaller payoff
to being a high-priced outlier. Our specification encompasses the polar settings of perfect
information and a standard search environment. In the perfect information environment,
consumers are armed with knowledge of the full vector of available prices. Our model collapses to a reduced form of this case if γ = 0 and α = 1, and we assign β = 1 to this situation.
By contrast, in a typical consumer search environment, consumers have an understanding
about the distribution of prices, without a knowledge of any specific firm’s price. Our specification could describe this case if γ = α = 1/n, and we assign β = 0 here.12 We collect this
discussion in the following Assumption.
Assumption 2. Market shares, s j (p, p, p), j = 1, . . . , k, are given by (1). Further, (i) γ(β) ∈
[0, 1/n] and α(β) ∈ [1/n, 1]; (ii) γ(0) = α(0) = 1/n; (iii) γ(1) = 0 and α(1) = 1; and (iv) γ0 (β) < 0
and α0 (β) > 0.
Profits for firm j are given by
π j (p, p, p) = R(p j )s j (p, p, p).
(2)
Each firm j simultaneously chooses a price p j ∈ A j = ℜ+ , with set of pure action profiles
Q
A = j A j . A pure strategy for firm j is a mapping from the set of all possible histories to the
set of actions. Let H t = A t be the set of t -period histories. The set of possible histories is then
S
t
H = ∞
t =0 H , and a strategy for firm j is a mapping σ j : H → A j . Given the strategy profile
t
σ, let a j (σ) be the period t action for firm j induced by σ, and let a t (σ) be the associated
action profile. Payoffs for firm j are then given by
V j (σ, p, p) =
∞
X
δt π j (a t (σ), p, p).
(3)
t =0
In equilibrium, no firm has an incentive to deviate given the strategies of her rivals and
given the market share relation (1). In addition, we impose a consistency requirement that
the tuple (p, p) is determined by the lower and upper bounds of the unconditional distribution induced by σ. Equlibrium is formalised with the following definition.
Definition 1. (1) A strategy profile σ is admissible with respect to (p, p) if p = inft , j a tj (σ) and
p = supt , j a tj (σ).
(2) A market equilibrium is a triple (σ, p, p) such that (i) σ is admissible with respect to (p, p),
and (ii) σ is a subgame perfect Nash equilibrium in the class of admissible strategies.
12
In this case, in the spirit of Diamond (1971), no Nash equilibrium to the single period counterpart of our
model exists with marginal cost pricing as long as p > 0. The existence of a monopoly pricing equilibrium will
depend on p and p.
7
This definition plays an important role. Admissibility requires that, on the equilibrium
path, consumer beliefs about the lower and upper bounds of the price distribution must be
correct. Effectively, consumers take as a reference the lowest and the highest prices they
have observed over a period of time. With the market share relation (1), this implies that
prices within this range are less effective at attracting the attention of consumers. Thus,
firms have to undercut the prices of their rivals outside the range of experience of consumers
if they wish to alert all consumers. Implicitly, we are assuming that consumers have some
awareness of the unconditional distribution of prices without being perfectly familiar with
the contemporaneous distribution.
Our interest is in optimal collusive strategies that are consistent with market equilibrium.
Firms choose strategies to maximise the objective function (3) subject to the usual incentive
compatibility constraints and our consistency requirement. Negotiating, monitoring, and
enforcing collusive policies may become more challenging for a cartel as the sophistication
of firm strategies grows. This may place an endogenous limit to the complexity of firm strategies. In this paper, we restrict firms to symmetric k-period sequences of prices on the equilibrium path. This restriction results in a tractable problem, amenable to analytic solution,
while still permitting us to consider interesting pricing dynamics. As we shall see, the cartel
has an incentive to raise the complexity of the price path in our model (i.e. permit higher k).
Hence, k could be determined as a trade-off between profitability on the equilibrium path
and the difficulty of coordinating a more complex path.
To simplify exposition, we define firm strategies in terms of revenues rather than prices.
This exposition is without loss of generality because: i) by Assumption 1, revenue is strictly
increasing in price up to the monopoly price; and ii) firms have no incentive to price above
the monopoly price. In addition, we normalise revenues so that R(p̂) = 1. This can be
achieved through suitable scaling of demand, D(p). Let πs be the revenue prescribed for
each firm in period s if firms follow strategies σk . We can then define the coordinated strategy σk as follows.
Definition 2. In a k-period cycle σk , each firm j implements the program:
(i) starting from period t = 0, infinitely repeat the sequence {π1 , π2 , . . . , πk } subject to πs ∈
[0, 1], s = 1, . . . , k;
(ii) in the event of a deviation by any player, henceforth set revenue equal to zero in every
period.
A special case of σk is a constant revenue path in which πs = π, s = 1, . . . , k, for constant
π. As a benchmark, the following Lemma handles this case.
Lemma 1. There exists a market equilibrium in which firms adopt a k-period cycle with πs =
π ∈ (0, 1] for s = 1, . . . , k if and only if δ ≥ (n − 1)/n.
Before we unveil the cartel’s problem, we introduce some notation. Define v s , s = 1, . . . , k
to be the continuation value for the cartel starting from period s of the strategy σk :
v1 =
π1 + δπ2 + · · · + δk−1 πk
1 − δk
, v2 =
π2 + δπ3 + · · · + δk−1 π1
1 − δk
8
, . . . , vk =
πk + δπ1 + · · · + δk−1 πk−1
1 − δk
,
and notice that
v 1 = δv 2 + π1 ,
v 2 = δv 3 + π2 ,
...,
v k = δv 1 + πk .
(4)
Assign πk to be the lowest point in the firm’s revenue sequence:
πk = min πs ,
s∈{1,...,k}
and let the cartel maximise v, defined as follows:
v = max[v 1 , . . . , v k ].
(5)
Notice that with v as the objective, assigning the minimum point of the cycle to position k
in the revenue sequence is without loss of generality. That is, it is equivalent to considering
v 1 as the cartel objective and allowing the position of the minimum price to be unrestricted.
We now set up the cartel’s problem through Lemma 2.
Lemma 2. Suppose δ < (n − 1)/n. Then the optimal k-period cycle σk consistent with market
equilibrium is given by the solution to the following program:
max
π1 ...πk ∈[0,1]
v
(6)
subject to
v s ≥ απs n, s = 1, . . . , k − 1, and v k ≥ πk n,
(7)
v s ≥ πk n, s = 1, . . . , k − 1,
(8)
v s ≥ γn, s = 1, . . . , k.
(9)
The cartel maximises the discounted value of the joint profit stream subject to three sets
of incentive compatibility constraints. To understand these constraints, first note that the
consumer price reference range [p, p] is associated with the revenue range [πk , 1]. πk is the
lowest revenue in the cycle. Applying admissibility then implies that R(p) = πk . Because the
cartel’s constraints are all homogeneous of degree one in revenues, p will be equal to the
monopoly price. Applying admissibility we have R(p) = 1.
There are three kinds of deviation the cartel must guard against. First, suppose firm j
considers deviating from cartel policies by marginally undercutting the current prescribed
price. For any period s < k of the cycle, this deviation is within the range [p, p]. Deviation
therefore yields market share α and profit απs . In period k, marginal undercutting is outside
the price bounds and is observed by all consumers, leading to profits πk . The constraints
in (7) therefore deter any deviation of this type. Second, with the market share function (1),
deeper price cuts do not yield a larger market share unless firm j sets a price below the lower
bound p. The constraints in (8) prevent this form of deviation. Third, firm j also considers
deviating by raising price. Any price less than or equal to p leads to market share γ, where
p is equal to the monopoly price. Therefore, the most profitable price rise involves setting
9
the monopoly price and earning revenue of 1. The constraints in (9) deter this deviation. For
future reference, number the constraints in (7) 1, . . . , k where constraint s involves v s .
With Lemma 2 in hand, we need only solve the program (6) - (9) to identify a market
equilibrium that is jointly optimal for firms. We consider non-trivial solutions with a strictly
positive cartel value, v > 0. The nature of the problem faced by the cartel depends on the nature of deviations that must be guarded against. In high salience environments (i.e. “high”
β environments), deviations are more often observed and recalled by consumers, and the
greatest danger to the cartel is the temptation to undercut. We deal with this case in Section
1.1 below. Alternatively, if salience is low, another threat to cartel discipline arises from the
temptation to raise price above the proscribed level, secure in the understanding that consumers are unlikely to notice that a deviator’s price is unusually high. We turn to this case in
Section 1.2.
1.1 High salience environments
In this Section we characterise the optimal cartel revenue path for parameter vector (β, δ, n, k)
if the salience parameter β is sufficiently high, as defined by Lemma 4 below. Our main results are contained in Lemma 3 where we solve for the optimal k-period cycle that is impervious to undercutting. The remaining results provide additional detail for our solution. In
Lemma 4 we show that, for sufficiently high β, the resulting revenue path also prevents deviations involving price rises. Lemma 5 demarcates ranges of values of the discount factor that
give rise to equilibria with different properties. Lemma 6 then provides an explicit solution
to the cartel’s optimal revenue path.
We first define the critical discount factor
αn − 1
δ1 (β, n, k) ≡
αn
µ
¶ k−1 µ
k
n −1
n
¶1
k
.
(10)
As we show in Lemma 3, δ1 is the lowest discount factor amenable to collusive equilibria in
high salience environments.
Lemma 3. The
cycle σk solves the program (6) - (8) if and only if δ ≥ δ1 . The cycle is
£ k-period
¢
unique. If δ ∈ δ1 , n−1
, the cycle has the following properties:
n
(i) revenues decline
monotonically
over the cycle: π1 ≥ nπ2 ≥ · · · > πok , with π1 = 1;
n
o
δv 1
δαn
δn
(ii) πs = min 1, αn−1
πs+1 , s = 2, . . . , k − 2, πk−1 = min 1, αn−1
πk , and πk = n−1
;
(iii) v = v 1 is an increasing function of δ.
An implication of Lemma 3 is that πs is increasing in δ for s = 1, . . . , k. This allows us to
offer the following definitions, which are useful for our discussion of the Lemma.
Definition 3. The knot discount factor δi connects two regions: if δ < δi the equilibrium
path has πi < 1; if δ ≥ δi the equilibrium path has πi = 1, for i = 2, . . . , k − 1.
Definition 4. i) An equilibrium price path is a pure sales path if πs = 1 > πk for s = 1, . . . , k −1.
ii) An equilibrium price path is a distinct cycle path if πs < 1 for s = 2, . . . , k − 1.
10
Lemma 3 implies that a dispersed price path is easier to sustain than a fixed price in high
salience environments. That is, there is a range of discount factors, [δ1 , (n − 1)/n), for which
collusion is sustainable with a dispersed price path, but not with a fixed price. In this range,
firms optimally coordinate on dispersed prices to reduce the visibility of any potential deviation, thereby relaxing the incentive constraints of the cartel. If β (and hence α) is smaller,
consumers have greater difficulty penetrating the haze of price dispersion and the cartel is
sustainable for a greater range of discount factors. Intuitively, if consumers have greater difficulty recalling specific prices that are within their realm of experience, then intertemporally
dispersed prices provide a greater shield against deviation by undercutting. Notice also that
in the limit, as consumer powers of observation and recall approach the ideal, the sustainability of collusive price paths converges towards the perfect information environment. That
is, δ1 converges towards (n − 1)/n as α approaches 1.
The optimal revenue path is described by a set of complementary slackness conditions.
If constraint s is not binding, firms obtain monopoly revenues; if it is binding, revenues must
decline to uphold the incentive constraints in (7). Because πk is the lowest revenue, a firm
undercutting in period k receives a market share of 1. Therefore constraint k is the most
difficult to satisfy, and it is always binding for δ < (n − 1)/n. For δ ∈ [δk−1 , δk ), constraint k
is the only binding constraint, and a pure sales path is observed. Sales obfuscate the price
process in the eyes of consumers by generating a range of observed prices, and achieve this
at minimum cost. That is, firms can charge the monopoly price in every period of the cycle
except the last.
For lower values of δ, πk must be reduced to satisfy the binding constraint k. Within
the region [δk−1 , δk ), the other constraints in (7) remain slack. At δ = δk−1 , constraint k − 1
becomes binding, and this constraint can only be satisfied in the face of a lower discount
factor by lowering πk−1 . As the discount factor is reduced below successive knot discount
factors, successive revenues must be lowered to satisfy the incentive constraints. In the limit
of this process, all revenues except the first in the sequence must be lowered. A distinct cycle
path occurs for δ ∈ [δ1 , δ2 ).
Our next Lemma demarcates the boundaries of high salience environments.
Lemma 4. Suppose {πs }ks=1 solve the program (6) -(8). Then there exists β1 (n, k) such that for
any β ≥ β1 , {πs }ks=1 also satisfy the constraints in (9). Specifically, β1 solves
γ(β1 ) = πk (β1 , δ1 (β1 )).
(11)
In the presence of price dispersion, the attentiveness of consumers has two opposing
effects on the profitability of deviation. For low values of β, consumers may not notice the
attractiveness of a low-priced firm, reducing the payoffs to undercutting the cartel price. At
the same time, a high-priced firm may still attract some custom, increasing the profitability
of deviations that involve price rises. Lemma 4 suggests that, for given values of n and k,
there is a critical value of β, above which raising price is never a profitable deviation.
The following two Lemmas complete our characterisation of equilibrium in high salience
environments. Lemma 5 describes the knot discount factors associated with the complementary slackness conditions in part (ii) of Lemma 3.
11
Lemma 5. The knot discount factors δs ∀ s ∈ [2, k − 1] are determined by the following equations:
¶
µ
¶
µ
X k−i
1 s−1
αn − 1 k−s n − 1
k
δs +
.
(12)
δ
=
αn i =1 s
αn
n
Part (ii) of Lemma 3 implies that the optimal revenue path is completely determined by
the revenue in the last period of the cycle, πk . We solve for this in Lemma 6. This permits an
explicit solution for the optimal revenue path and accompanying value of the cartel’s objective function.
Lemma 6. For s = 1, . . . , k − 1, if δ ∈ [δs , δs+1 ), then a k-period cycle solving the program (6) (8) yields
(n − 1)(1 − δs )
´
,
v1 = ³ ³
¡ αn ¢k−s−1 ´
n 1 − αn−1
δk − 1 (1 − δ)
δ(1 − δs )
´
πk = ³ ³
.
¡ αn ¢k−s−1 ´
n 1 − αn−1
δk − 1 (1 − δ)
(13)
Together, Lemmas 3 - 6 constitute a complete description of equilibrium in high salience
environments. Example 1 illustrates the equilibrium and provides a motivation for the market share equation (1).
Example 1. Consider the following setting. A fraction β of consumers are shoppers.13 These
consumers are perfectly informed about prices and buy from the cheapest firm available.
The remaining 1 − β non-shoppers have limited price awareness. Only unusual prices attract
their attention. They are perfectly informed about any prices that are outside their range of
experience, but they are unable to recall any specific price similar to those they have previously observed. In particular, if there are any prices with p < p, then non-shoppers purchase at the lowest available price. If no firms choose a price p ∉ [p, p], then no specific
price attracts the attention of these consumers and they purchase from each firm with equal
probability 1/n. This leads to market shares of the form (1) with
γ(β) =
1−β
,
n
α(β) = β +
1−β 1
n −1
= +β
.
n
n
n
A larger proportion of shoppers leads to a greater payoff from undercutting the prices of
your rivals, and a smaller payoff from setting a higher price than your rivals.
We consider the following parameters. There are n = 5 firms who pursue strategies with
cycle length k = 4, and a fraction β = 0.15 of consumers are shoppers. Figure 1 depicts the
resulting equilibrium revenue path for a range of possible discount factors. The vertical axis
describes revenue and the horizontal axis indexes discount factors. For a given discount
factor, we can read the picture vertically to reveal revenues at each point of the cycle. Recall
that π1 = 1 and that each knot discount factor δs delineates regions where πs = 1 and πs < 1.
13
In this example, consumers are heterogeneous. This is not required for our results. For example, in the
consumer search model of Appendix B, consumers are ex ante identical.
12
1
π1
π2
Revenue
π3
π4
γ
0
0.4
δ 1 δ2
δ3
Discount factor
δ4 = 0.8
Figure 1: Example 1
The knot discount factors are indicated with δ1 ≈ 0.453, δ2 ≈ 0.469, δ3 ≈ 0.502, and δ4 = 0.8.
For δ > δ4 = 0.8, a constant revenue path is sustainable. A price cycle equilibrium exists for
δ ∈ (δ1 , δ4 ). For δ ∈ [δ3 , δ4 ) we observe a pure sales path. As δ falls, π4 falls, until we hit the
knot discount factor δ3 , at which point, π3 begins falling as well. For δ ∈ [δ2 , δ3 ) we then
observe more extensive sales with two periods of monopoly pricing in each cycle and two
periods of discounted revenues in between. Finally, in the region δ ∈ [δ1 , δ2 ), we observe
distinct cycle paths. Note that γ = 0.17 is always less than π4 , indicating that β > β1 and the
constraints in (9) play no role in this example.
Figure 2 illustrates the path of revenues arising from Example 1 for different discount factors. In each panel, revenues are shown on the vertical axis and the horizontal axis indexes
time. δ = δ1 is the lowest discount factor conducive to a collusive equilibrium, and this case
is shown in the top left panel. We observe a distinct cycle path, with revenues varying between 1 and approximately 0.18 in an asymmetric cycle. With the higher discount factor of
δ = δ2 shown in the top right panel, a more profitable revenue sequence is sustainable. We
can see that constraint 2 is now slack, permitting a 2-period spell of monopoly pricing, and a
slightly higher cycle minimum of π4 ≈ 0.2. The bottom left panel shows an example of a sales
path with δ = δ3 . A regular sale is observed every 4th period. As the discount factor is raised
further, the depth of the sale is reduced. In the limit (that is for δ ≥ δ4 ), no sale is required to
maintain the incentives for cooperation, and a constant revenue path is observed. This case
is shown in the bottom right panel.
13
1
0
Revenue
Revenue
1
δ = δ1
Time
0
δ = δ2
Time
Revenue
1
Revenue
1
0
δ = δ3
Time
0
δ = δ4
Time
Figure 2: Price cycles
1.2 General salience environments
In Section 1.1, we solved the cartel’s problem under the presumption that the only threat to
cartel discipline came from the temptation to undercut. When salience is low, the temptation to deviate by raising price may also be a threat. A firm setting a higher price than her
rivals loses fewer customers if consumers are less attentive. This increases the incentive to
raise price, particularly when cartel policies call for a low price. Our first result summarises
the impact on the cartel.
Lemma 7. Suppose the revenues {πs }ks=1 solve the program (6) - (8) and suppose β < β1 (n, k)
where β1 is defined in (11). Then, there exists δ̂(β, n, k) such that for δ < δ̂, there is no solution
to the program (6) - (9) with positive revenues, while for δ ≥ δ̂, {πs }ks=1 also solve the program
(6) - (9). In particular, δ̂ solves
γ(β) = πk (β, δ̂).
(14)
Recall from Lemma 4 that if consumers are sufficiently attentive, the temptation for firms
to relent is overshadowed by the desire to undercut. The constraints in (9) can then be ignored. Lemma 7 refines this result by describing the consequences if consumers are inattentive. In this case, the constraints in (9) determine the sustainability of collusion. If firms
are sufficiently patient, collusive profits can still be salvaged. Further, if these constraints are
satisfied, the optimal revenue path is entirely unaffected by them.
14
This result brings into focus the contrasting roles of the three sets of incentive compatibility constraints. The constraints in (7) prevent marginal undercutting. These constraints
determine the shape of the revenue path and, in high salience settings, also determine the
sustainability of the cartel. The constraints in (8) ensure firms have no incentive to undercut below the consumer reference price p. These constraints are always satisfied when (7)
holds. We can see this by comparing the two sets of constraints. An optimal revenue path
that satisfies (7) yields a cartel continuation value that decreases monotonically over time
within each price sequence. Therefore, if constraint k in (7) is satisfied, then all constraints
in (8) will also hold.
Finally, the relenting constraints in (9) play a binary role. In markets with attentive consumers, these constraints can be ignored. With inattentive consumers, these constraints
determine the sustainability of collusion, but have no effect on the shape of the optimal revenue path if collusion is sustainable. This last result arises from a direct conflict between
the two sets of constraints (7) and (9). To ease concerns about undercutting, the cartel must
lower revenues to reduce the current payoffs to deviation. By contrast, to mitigate the incentive to relent, the cartel must raise revenues in order to raise the continuation value of the
cartel relative to the monopoly profit that a deviator would receive. An optimal cartel seeks
to maximise cartel value. Therefore, revenues will be maximised subject to the marginal
undercutting constraints in (7). If the constraints in (9) are also satisfied, then collusion is
feasible. If they are not, then there is no further scope to raise revenues.
The following example illustrates the determination of the optimal revenue path in lowsalience environments.
Example 2. Reconsider the setting of Example 1, and suppose there are n = 5 firms who
choose an optimal cycle of length k = 4, and a measure β = 0.075 of consumers are shoppers. Notice that the only change we have made to Example 1 is to reduce the measure of
shoppers. Figure 3 depicts the resulting equilibrium revenue path for a range of possible discount factors. The vertical axis describes revenue and the horizontal axis indexes discount
factors. For a given discount factor, we read the picture vertically to show revenues at each
point of the cycle. In this example, γ ≈ 0.185 is not always less than π4 . The critical condition γ(β) = π4 (β, δ) results in δ = δ̂ ≈ 0.435. Consequently, the equilibrium exists only for
δ ∈ (δ̂, δ4 ). For values of δ in this range, equilibrium is unaffected by the constraints in (9).
Note also that δ3 = 0.389 < δ̂, which means only the equilibrium with pure sales is feasible.
Lemma 7 implies that collusion requires cartel vigilance towards both undercutting and
relenting. A sufficiently patient cartel can overcome both obstacles. For β ≥ β1 , Lemmas 3
and 4 demonstrate that collusion is sustainable if and only if δ ≥ δ1 . For β < β1 , collusion
is sustainable for δ ≥ δ̂ where δ̂ is determined by (14). Combining these results, collusion is
sustainable if and only if δ ≥ δ∗ where the critical discount factor δ∗ is defined as
(
δ1 (β, n, k) if β ≥ β1
δ∗ (β, n, k) =
(15)
δ̂(β, n, k)
if β < β1 .
15
1
π1
π2
Revenue
π3
π4
γ
0
0.3
δ̂
Discount factor
0.8
Figure 3: Example 2
Following on from our discussion of Lemma 3, let us now consider comparative statics
with respect to β in low salience environments. By Assumption 2, as β converges to 0, α
approaches 1/n; with extreme consumer inattentiveness, the market share benefits of undercutting are minimal. This is reflected in δ1 , which converges to 0 as α converges to 1/n. If
we were to disregard the constraints in (9), this suggests that a k-period cycle would always
be sustainable. However, because of the threat of deviation by raising price, collusion may
not be sustainable for low values of β. The constraint δ ≥ δ̂ applies in these cases.
The salience parameter β also has implications for the shape of the revenue path. The
following result demarcates values of β for which alternative shapes are optimal.
Lemma 8. Suppose the revenues {πi }ki=1 solve the program (6) - (8). Then, for s = 2, . . . , k − 1,
there exists βs (n, k) such that if β > βs , the constraints in (9) are satisfied if δ ∈ [δs , δs+1 ).
Recall that the knot discount factors δs determine the shape of the cycle in terms of the
number of distinct revenues observed on the equilibrium path. Lemma 8 places restrictions
on which of these cycle shapes could be optimal depending on the attentiveness parameter
β. Taking Lemmas 4, 7 and 8 together, we can summarise the effect of the constraints (9) on
the cartel’s program for different salience parameters. If salience is sufficiently high (β > β1 ),
then these constraints are always satisfied, and we can rely on the solution we discussed in
Section 1.1. For β ∈ (β2 , β1 ), there is a range of discount factors for which the distinct cycle
equilibria are impacted by (9). At β ≤ β2 , no distinct cycle paths satisfy these constraints
16
optimally. For β ≤ βs , all optimal paths involve at most k − s + 1 distinct revenues. For β ≤
βk−1 , the only optimal equilibria that survive these constraints are sales paths.
The easiest way to see the way this process works is to re-examine Examples 1 and 2 as
depicted in Figures 1 and 3, respectively. β1 is determined by the intersection of γ and π4 . In
Example 1, γ intersects π4 for δ < δ1 . For all δ, equilibria are unaffected by the constraints (9).
In Example 2, γ intersects π4 for δ > δ3 . No equilibria to the program (6) - (9) exist for δ < δ̂.
The only optimal equilibria that remain (for δ > δ̂) involve pure sales paths.
We illustrate the manner in which β influences cartel sustainability and the shape of the
revenue path with the following example.
Example 3. Reconsider the setting of Example 1 with n = 5 and k = 4, and consider all β ∈
[0, 1]. Figure 4 illustrates the determination of critical discount factors. The vertical axis
depicts the discount factor and the horizontal axis indexes the salience parameter β. We
can read this figure vertically. Fixing a particular value of β determines the knot discount
factors δs and the critical value of the discount factor. Equilibrium exists if and only if δ ≥
δ∗ as defined in (15). If β ≥ β1 , then δ ≥ δ1 is required and cartel sustainability becomes
harder for higher values of β. That is, δ1 is increasing in β and δ1 converges to (n − 1)/n as
β approaches 1. For β < β1 , relenting is a concern for the cartel and sustainability requires
δ ≥ δ̂. We observe a non-monotonic relationship between δ̂ and β and for low values of β,
collusion becomes harder to sustain for lower values of β.14
Notice also that for low values of β, all optimal paths involve sales. In particular, this
occurs for β below β3 , corresponding to the intersection of δ̂ and δ3 . On the other hand, for
large values of β, other dynamic structures may be optimal, including distinct cycle paths. We close this section by collecting our main results into the following two propositions.
Proposition 1 discusses the requirements for existence of a non-trivial collusive equilibrium.
Proposition 2 characterises the resultant equilibria.
Proposition 1. There exists a market equilibrium with positive payoffs using strategies σk if
and only if δ ≥ δ∗ (β, n, k) as defined in (10), (14), and (15).
Proposition 2. An optimal market equilibrium with the strategies σk has the following revenue path:
(i) if δ ≥ (n − 1)/n, then πs = 1 for s = 1, . . . , k;
o
n
o
n
δαn
δn
πk , πs = min 1, αn−1
πs+1
(ii) if δ∗ ≤ δ < (n−1)/n, then πk is given by (13), πk−1 = min 1, αn−1
for s = 2, . . . , k − 2, and π1 = 1.
14
To see that δ̂ may not be monotonic in β, consider two cases. In the case of pure sales, the right hand side
of (14) is independent of β. This is because only constraint k in (7) is binding, and constraint k does not depend
on β. The left hand side of (14) is decreasing in β, and hence we observe an inverse relationship between δ̂ and
β. By contrast, in the case of a distinct cycle, all k constraints in (7) are binding. Examining these constraints
reveals an inverse relationship between revenues and β. The right hand side of (14) is then also decreasing in
β, and we could observe a positive relationship between δ̂ and β.
17
0.8
δ4=0.8
δ3
δ2
δ1
δ
∧
δ
0
β
β3 β2 β
1
1
Figure 4: Example 3
1.3 Cycle complexity
Equilibrium is determined by the set of parameters (n, k, δ, β). Of these parameters, the
number of firms, their patience, and consumer attentiveness might be beyond the complete control of the cartel. However, the length of the cycle may be a choice variable, and
we examine this choice in the next Proposition.
Proposition 3. The optimal k + 1 period cycle always dominates any cycle with k periods.
Absent costs of coordination, firms prefer a more complex price path. Intuitively, a longer
sequence affords the flexibility to have a more profitable initial phase in each sequence,
while maintaining a substantial range of prices to dampen the powers of observation of consumers. By increasing the value of the cartel, a more complex price path relaxes the incentive
compatibility constraints, making collusion sustainable for a greater range of discount factors.15 If coordination is costly, we might then expect the trade-off between complexity and
profitability to yield an optimal finite cycle length.
15
For the case of β ≥ β1 , this is easy to see. From equation (10), δ1 , the minimum discount factor consistent
with collusive equilibrium, is strictly decreasing in k.
18
2 Discussion
2.1 Consumer beliefs
Subgame perfection is silent on the beliefs of players both on and off the equilibrium path.
Admissibility introduces an equilibrium refinement that imposes consistency between consumer beliefs and firm behaviour on the equilibrium path. In this section, we briefly discuss
the implications of this refinement and informally consider an alternative refinement.
Definition 1 determines the information that acts as a reference to trigger the attentiveness of consumers. Consumers are spurred to retain and recall pricing information that is
unusual with respect to their entire history of observation. Effectively, in deciding which
prices are unusual or important to her, each consumer takes as a reference the prices she
is accustomed to seeing over a sustained period of observation.16 Thus, she is familiar with
the unconditional distribution of prices, but she does not know the contemporaneous distribution. Our intent with this specification is to explore the implications of the consumer’s
incomplete understanding of the dynamic process generating prices rather than to suggest
that consumers are completely naive of this process. The optimality of intertemporal price
dispersion follows intuitively. A dispersed price path clouds each consumer’s understanding
of the price distribution. For a deviation to be fruitful, more dramatic price cuts are needed
to attract the hazy gaze of consumers, thereby relaxing the incentive constraints of the cartel.
Definition 1 does not impose consistency between consumer beliefs and firm behaviour
off the equilibrium path. In the event of a deviation in which firm j chooses a price p j ∉
[p, p], the bounds p and p do not adjust. If p > 0, a punishment regime with zero value is
then available. Specifically, all firms set price equal to marginal cost, consumers notice these
unusually low prices, and there is no profitable deviation.
Let us briefly consider an alternative refinement in which p and p are determined by
the lowest and highest prices observed on or off the equilibrium path. Effectively, the fog
of obfuscation extends to encompass prices observed in punishment. For a given strategy
profile, this adjustment to our model has no effect on the value of play on the equilibrium
path. However, it does alter the value of punishment in the event of deviation. To see this,
suppose punishment involves coordinated marginal cost pricing. The lower bound would
then adjust to p = 0. Then, in contrast to the model of Section 1, there is a profitable deviation involving a price p > p. This alters the cartel’s basic trade-off by reducing the costs of
deviation. We have used simulation methods to examine this alternative refinement.17 Our
main conclusions are preserved with this specification. In particular, for the simulations that
we have considered, a price cycle is sustainable for a greater range of discount factors than a
fixed price, and the structure of the optimal revenue path remains intact.
16
An alternative interpretation consistent with our definition of cartel strategies is that consumers take as a
reference the prevailing prices over the last k periods.
17
Note that this adjustment to our model also necessitates the construction of optimal penal codes in the
manner of Abreu (1988). We perform this task for a more general setting in Appendix C.
19
2.2 Market shares
In this section, we briefly consider several properties of the market share function (1). First,
market shares are discontinuous at the price bounds p and p. Discontinuity is not important for our results. For example, suppose that market shares transition linearly between 1
and α in the neighbourhood [p, p + ²].18 Then we can show that market equilibrium is unaffected as long as the transition is sufficiently fast (i.e. ² is sufficiently small). Minor changes
in the market share function of a more general nature will not change the character of the
optimal revenue path. This is because the optimal revenue path is primarily determined by
discounting. Changes in the market share function may affect the most tempting deviation
from cartel policies without substantially impacting on equilibrium play.
Second, the market share function specifies that any price within the bounds p and p
is equally salient to consumers, leading to constants α and γ. In practice, the observability of a price may depend either on the price itself or the empirical distribution of prices
observed by consumers. For example, consumers may pay more attention to lower prices
within these bounds. Alternatively, consumers may pay more attention to rarely observed
prices in a more graduated manner than we have specified. It is our conjecture that extending our model in this manner will alter the sustainability of pure sales paths relative to distinct cycle paths. In a pure sales path, only two prices are observed on the equilibrium path,
while a range of prices are observed in a distinct cycle path. Hence, a distinct cycle path may
be more confusing to some consumers, increasing its sustainability relative to a pure sales
path for some alternative specifications of (1).
Third, in Appendix C we adopt a more general approach by imposing relatively mild restrictions on firm profit functions and describing the conditions under which price dispersion leads to the improved sustainability of collusion. This approach allows us to highlight a
potential trade-off between the role of price dispersion in reducing the payoffs to deviation
and a possible role in influencing the effectiveness of punishment. There is no such trade-off
in Section 1: a zero-value punishment subgame is available whether strategies call on firms
to adopt a fixed price or a price cycle on the equilibrium path. However, it is conceivable
that the propensity of consumers to observe prices in a price war could be impacted by the
variability of prices on the equilibrium path.
We expect the effect on the value of punishment to be minor. A price war would represent a substantive break from the pricing pattern observed on the equilibrium path, with
the likely effect of shifting consumers’ attention towards the price war. The nature of pricing
dynamics on the equilibrium path may therefore have a minor impact on the profitability of
punishment.
2.3 Collusion with a competitive fringe
In many market settings, a cartel does battle with firms outside its reach. Suppose a single
myopic firm competes alongside the cartel. The implications for the sustainability of a fixed
18
For illustration, we consider the discontinuity at p. Similar arguments apply at the other discontinuities
in the market share function.
20
price are dramatic. If market shares are determined by (1), there is no equilibrium in which
the cartel maintains a fixed price; the competitive fringe could capture the whole market by
marginally undercutting the cartel price. By contrast, if firms are sufficiently patient, nontrivial equilibria with dispersed prices may be sustainable. To see this, note that it is only
at the bottom of a price cycle that the fringe can capture the entire market by marginally
undercutting. At other points in the cycle, the cartel could still be profitable even with an
undercutting competitor.
A competitive fringe introduces additional strategic considerations for the cartel. Principally, the fringe must be discouraged from undercutting below p. A substantial sale is required in order to minimise the payoff to the fringe from undercutting. Consequently, even
if cartel members are very patient, substantial price variation is observed on the equilibrium
path.
2.4 Asymmetric equilibria
We have restricted attention to symmetric equilibria of the repeated game. Asymmetric equilibria will be more profitable in many settings. Consider, for example, a pure sales path. An
asymmetric pure sales path in which a single firm has a sale every k periods will achieve the
same obfuscation properties in our model. This can be seen with reference to Definition 1.
Such a path may be jointly more profitable for the cartel as a single firm bears the burden of
obfuscation. Two factors might limit the applicability of such asymmetric paths. First, they
introduce additional complexity to the coordination problem of firms as they also require coordination on the identity of the low-priced firm each sale period. This introduces a trade-off
between profitability and complexity reminiscent of the choice of the cycle length k. Second, such asymmetric paths introduce contemporaneous price dispersion. This presents an
incentive for consumers to pay greater attention to prices by providing contemporaneous
search benefits.
2.5 Pricing dynamics and market characteristics
In our model, the obfuscation properties of a price path are summarised by the price range
[p, p] and the attentiveness parameter β. In markets requiring frequent purchases, consumers may develop a more refined understanding of the price distribution. Markets with
highly visible prices may also foster consumer price awareness. We can capture such variation across markets by adjusting β.
In a more general framework, the power to obfuscate may depend on other properties of
the price path such as the complexity of the path or the number of distinct prices in a cycle. More complicated paths may achieve greater obfuscation. For example, pure sales paths
may be more transparent to consumers than distinct cycle paths. This opens up the possibility that firms may be able to overcome the attentiveness of consumers through greater
efforts at obfuscation. We could see more intricate price paths in market settings with highly
visible prices and frequent purchases.
21
We illustrate this point with two examples. In the market for retail petrol, consumers purchase frequently and prices are prominently displayed on billboards that are easily observed
while driving. Greater efforts may therefore be needed to confound price comparison. Accordingly, many retail petrol markets exhibit price cycles that resemble our distinct cycle
paths. Further, pricing dynamics are quite well synchronised between firms. Such paths are
quite opaque to consumers while also minimising the contemporaneous benefits of consumer search.
The market for retail groceries is markedly different. When consumers decide where to
purchase groceries, the price comparison exercise is a complicated one. Many products are
included in the grocery bundle, and the price of each product may not be surveyed on each
shopping excursion. In addition, only a single retailer may be typically observed at a time.
In these circumstances, only modest efforts may be required from retailers to obfuscate the
consumer price comparison exercise. Occasional sales for a subset of the product space may
be sufficient to hamper consumers in these efforts. Moreover, such sales may not need to be
synchronised between firms.
3 Conclusions
In this paper, we examined cartel behaviour in the presence of imperfectly observant consumers. Intertemporal price dispersion emerges as a mechanism to exploit consumer inattentiveness. The mechanism is quite intuitive. Price dispersion makes it harder for consumers to understand the price distribution, hampering their ability to detect deviations
from proscribed cartel policies. This reduces the payoffs to deviation, making collusion easier to sustain.
Our model nests the perfect information environment, and in this setting, a fixed price
is optimal for the cartel, but is sustainable for a relatively limited range of discount factors.
Once we relax the perfect information assumption, the optimal price path involves a sequence of monopoly prices interspersed with sales. For lower values of the discount factor,
the optimal price path must adjust to maintain the incentives of the cartel. In particular,
sales must be deeper, and extend for greater length for lower discount factors. In the limit,
we find asymmetric price cycles that resemble Edgeworth cycles provide the greatest protection against the impatience of firms.
We also examine comparative statics with respect to consumer attentiveness. As attentiveness converges to the ideal, we approach the perfect information environment: the range
of discount factors for which collusion is sustainable collapses towards the perfect information case. For high values of attentiveness, the greatest threat to cartel stability comes from
the prospect of undercutting cartel prices. Here, lower values of attentiveness increase the
effectiveness of dispersed prices as an obfuscation device, making collusion easier to sustain
relative to a fixed price. For low values of attentiveness, deviations involving price rises are
also a pressing concern. In this case, lower values of attentiveness may not aid cartel stability.
22
Appendices
A Proofs
Proof of Lemma 1
Proof. Suppose the k-period cycle σk proscribes the constant revenue path πs = π ∈ (0, 1], s =
1, . . . , k. Then p = p. A deviation involving a higher price attracts a market share of 0 and is
unprofitable. Deviation by undercutting yields a market share of 1. The strategy σk gives
value π/(n(1 − δ)), while deviation by marginally undercutting yields payoff approaching π.
This deviation is unprofitable if and only if δ ≥ n−1
.
n
Proof of Lemma 2
Proof. First, we establish that infinite reversion to marginal cost pricing is an optimal penal
code. Note that for a non-trivial solution to the program we must have πk > 0. Otherwise, we
could increase πk and hence increase v, without violating any of the constraints in (7), (8),
or (9). Hence, we must have p > 0. Now consider the stage game of our model consisting of a
single period of play. Assumption 2 implies that there exists a Nash equilibrium to the stage
game in which each firm sets a price of zero. To see this, notice that if p > 0, then any firm
j setting price p j > 0 will obtain market share of zero if her competitors set a price of zero.
Therefore, there is a Nash equilibrium to the stage game in which firms price at marginal
cost and obtain zero profits, and infinite reversion to marginal cost pricing is an optimal
penal code.
Second, note that a standard application of the one-shot deviation principle is possible.
See, for example, Fudenberg and Tirole (1991). We restrict attention to one-shot deviations
below. Let us show that if δ < (n − 1)/n then a constant revenue path in which πs = π, s =
1, 2, . . . , k for constant π is not sustainable. If σk proscribes a constant revenue path, then
p = p and any firm undercutting the price of her rivals will receive a market share of 1. The
strategy σk yields value π/(n(1−δ)), while deviation by marginally undercutting yields payoff
approaching π. If δ < (n − 1)/n, this deviation is profitable.
Third, we show that the prevention of marginal undercutting leads to the constraints
in (7). Suppose following history h t , σk calls on players to choose revenue πs with πs ≥
πk . Then the continuation value induced by σk following h t is v s /n. If πs > πk , a marginal
undercut of rival prices yields current revenue marginally below πs per customer and market
share α. The restriction v s ≥ απs n is therefore sufficient to deter this deviation. If instead
πs = πk , undercutting leads to market share 1 rather than α, and this case is dealt with in the
constraints (8). Similarly, marginal undercutting of the price associated with πk leads to a
market share of 1, leading to the last constraint in (7).
Fourth, we show that no other deviations involving undercutting the current price vector
are possible if the constraints in (8) are satisfied. Suppose σk calls on πs to be played in the
current period with πs > πk . Any deviation π0 ∈ [πk , πs ) will yield market share α, making a
23
marginal undercut the most profitable deviation. Alternatively, undercutting below πk will
yield market share of 1. This deviation is not profitable if v s ≥ πk n, leading to the constraints
in (8).
Finally, the only other possible deviation to consider is a price rise. Any price above p will
be noticed by all consumers leading to a zero market share. The optimal price rise therefore
yields revenue per customer of maxi πi , giving associated market share of γ. This deviation is never profitable if v s ≥ γ maxi πi n. Notice that this, and our other constraints, are
all homogeneous of degree one with respect to revenues. Let us prove by contradiction that
maxi πi = 1. By our normalisation of revenues, πi ≤ 1 ∀ i . Suppose that maxi πi < 1. Introduce the variables π0s = πs / maxi πi ∀ s ≤ k. Because of first degree homogeneity, our
transformed variables must satisfy the constraints. Given that π0s > πs ∀ i , it means that
v 0 > v. Consequently, there is a contradiction and maxi πi = 1. The constraints in (9) therefore ensure that there is no incentive to raise prices.
Proof of Lemma 3
Proof. 1) Initially, disregard the constraints in (8). Once we have described the equilibrium,
we show that in equilibrium these constraints are satisfied. Number the constraints in (7)
1, . . . , k, where constraint s involves v s .
.
2) As a preliminary step let us prove that the multi-price equilibrium exists only if δ > αn−1
αn
Add up all constraints in (7) to derive
((π1 + π2 + · · · + πk )(1 + δ + · · · + δk−1 ))/(1 − δk ) ≥ αn(π1 + π2 + · · · + πk ) + (1 − α)nπk .
Simplify the above expression
(π1 + π2 + · · · + πk ) (1/(1 − δ) − αn) ≥ (1 − α)nπk .
1
> αn, which means that δ > αn−1
Given that prices are positive it follows that 1−δ
αn .
3) Show that for each s = 1, . . . , k, constraint s is either binding or πs = 1. Consider constraint
s and suppose otherwise that the constraint is not binding and that πs < 1. By increasing πs
until either πs = 1 or constraint s is binding, the objective function v is increased and all the
constraints in (7) are satisfied, leading to a contradiction.
4) Observe that constraint k must be binding. Suppose otherwise that πk = 1. Then (4)
and the last constraint imply v k = δv 1 + 1 ≥ n. With δ < n−1
n , this means that v ≥ v 1 > n.
1
Notice that setting revenue πs = 1 for all s yields value 1−δ
< n, leading to a contradiction.
Consequently, πk < 1 and the last constraint is satisfied with equality, v k = πk n. Using (4),
δv 1
note that πk = n−1
.
5) Next, we show that for each s = 1, . . . , k, if πs = 1 then πs−1 = 1. If πs = 1 then constraint
s becomes v s ≥ αn. Constraint s − 1 stipulates v s−1 ≥ απs−1 n. Using equality v s−1 = δv s +
αn−1
αn−1
πs−1 from (4), transform this to v s ≥ αn−1
δ πs−1 . With δ > αn , this implies that αn > δ .
Consequently πs−1 = 1 satisfies constraint
s −o1.
n
δn
6) We now show that πk−1 = min 1, αn−1
πk . Using (4) to transform constraint k − 1, we
obtain δv k + πk−1 ≥ απk−1 n. Observing that constraint k is binding and rearranging, we
δn
obtain πk−1 ≤ αn−1
πk . Applying point 3) above yields our desired result.
24
n
o
δαn
7) Similarly, we show that πs = min 1, αn−1
πs+1 , s = 2, . . . , k − 2. First, note that if constraint
s + 1 is not binding then πs = 1. Alternatively, suppose constraint s + 1 is binding. We can use
(4) to transform constraint s to obtain δv s+1 + πs ≥ απs n. Using the fact that constraint s + 1
δαn
is binding, we obtain δαnπs+1 + πs ≥ απs n or πs ≤ αn−1
πs+1 . Applying point 3) above yields
our desired result.
imply π1 ≥
8) Let us prove that v = v 1 . Note that points 6) to 7) and the fact that δ > αn−1
αn
π2 ≥ · · · ≥ πk−1 > πk ; this result follows immediately.
9) We now show that π1 = 1. Suppose otherwise that π1 < 1. Notice that in point 8) it is proved
that revenues are monotonically declining over the cycle. This implies that πs ≤ π1 < 1 ∀ s,
and hence all constraints are binding. Now introduce new variables π0i = πi /π1 ∀ i ≤ k.
These new variables have to satisfy the constraints, because the original variables do and
constraints are homogeneous of degree one with respect to revenues. Given that π0i > πi
∀ i , it means that v 0 > v. Consequently, there is a contradiction and π1 = 1.
10) Next, we show that in equilibrium the constraints in (8) are satisfied. Let us consider
δn
πk .
two separate cases: πk−1 < 1 and πk−1 = 1. If πk−1 < 1 then from point 6) πk−1 = αn−1
This implies that απk−1 > πk . Combined with the fact that π1 ≥ π2 ≥ · · · ≥ πk−1 and the
constraints in (7), this is enough to prove that constraints in (8) hold. If πk−1 = 1 then using
5), π1 = π2 = · · · = πk−1 = 1. That means v k < v s for any s. Combined with the constraint k in
(7), this is enough to prove that the constraints in (8) hold.
11) Let us show that ∀ δ, the optimal sequence {πi }ki=1 , if it exists, must be unique. Assume
the opposite that there are two sequences. v 1 must be the same for both sequences, otherwise the one with the higher v 1 is chosen. This will uniquely determine the value of πk and
then, recursively, πk−1 ,. . . , π1 . The optimal sequence is therefore unique.
12) Next, derive δ1 . Consider the situation when all constraints are binding. From constraint
v
. Continuing this
k − 1 it follows that v k−1 = απk−1 n. Using (4) it follows that v k = αn−1
δαn k−1
process results in
αn − 1
vk =
δαn
µ
¶k−i
vi ,
i = 1, . . . , k − 1.
(16)
¡
¢k−1
In particular, v k = αn−1
v 1 . Then, using (4) and v k = πk n from constraint k, we obtain
δαn
¡ αn−1 ¢ k−1 ¡ n−1 ¢ 1
k
.
δ1 = αn k
n
13) Next, note that v 1 (δ) is strictly increasing in δ, where v 1 (δ) is the value of v 1 associated
with the optimal revenue sequence for discount factor δ. Consider any two discount factors
δ and δ0 with δ0 > δ. Let π = {πi }ki=1 be the optimal revenue sequence associated with δ
and π0 be the corresponding sequence for δ0 . Then, abusing notation slightly, v 1 (δ0 , π0 ) >
v 1 (δ0 , π) > v 1 (δ, π), as required.
14) Finally, show that a multi-price equilibrium exists if and only if δ1 ≤ δ < n−1
n . Recall that
when δ = δ1 , constraint 1 in (7) is binding and v 1 = αn. Because v 1 is strictly increasing in δ,
if δ < δ1 then v 1 < αn, leading to violation of constraint 1. Similarly, if δ > δ1 then v 1 > αn
and all constraints are satisfied.
25
Proof of Lemma 4
Proof. Comparing constraints (8) and (9), we see that when γ ≤ πk the constructed equilibδv 1
rium is not affected by the constraints (9). From Lemma 3, πk = n−1
. The lowest possible
value of πk consistent with equilibrium occurs when v 1 is at a minimum, suggesting that
constraint 1 in (7) is binding. Observing that v 1 = αn yields πk = δαn
n−1 . This leads to the inequality δ ≥
γ(n−1)
αn .
Given that equilibrium does not exist if δ < δ1 , the equilibrium is not
γ(n−1)
affected by constraints (9) when δ1 ≥ αn . Simplifying this expression, we observe that
µ 1 ¶ k−1
k
n−
. By Assumption 2, γ(0) = α(0) = 1/n. This expression is therefore violated
γ ≤ α n−1α
when β = 0. By contrast, γ(1) = 0 and α(1) = 1, and the expression is satisfied for β = 1. Noting that γ0 (β) < 0 and α0 (β) > 0, the left-hand side of the expression is strictly decreasing in
β, while the right-hand side is strictly increasing. Therefore, there exists a β1 such that for
any β ≥ β1 , {πs }ks=1 also satisfy the constraints in (9).
Proof of Lemma 5
Proof. At the knot discount factor δs , πs = 1, and constraint s is binding so that v s = αn.
¡
¢k−s n−1 αn
v (n−1)
Using equation (16), we can then show v 1 = k nδ = αn−1
αn
n δk−s+1 . On the other hand,
recursively employing (4), v 1 can be represented as v 1 = δv 2 +1 = · · · = δs−1 v i +δs−2 +· · ·+1 =
αnδs−1 + δs−2 + · · · + 1. Combining both relationships gives
µ
¶
µ
¶
X k−i
αn − 1 k−s n − 1
1 s−1
δ
=
.
δ +
αn i =1
αn
n
k
The lemma therefore is proved.
Proof of Lemma 6
Proof. First let s = k −1, that is consider the range δ ∈ [δk−1 , δk ) and recall that there is a sales
(n−1)πk
path in this range. Using (4), note that v 1 =
. Also, applying the definition of v 1 directly
δ
1+δ+···+δk−2 +δk−1 πk
.
1−δk
k−1
(n−1)(1−δ
)
.
(n(1−δk )−1)(1−δ)
gives v 1 =
Combining these together results in πk =
δ(1−δk−1 )
(n(1−δk )−1)(1−δ)
and
v1 =
Now using a similar approach let us prove the statement for any s = 1, . . . , k − 2. That is,
(n−1)πk
consider the range δ ∈ [δs , δs+1 ). In this case condition v 1 =
still holds, while the secδ
1+δ+···+δs−1 +δs π
+···+δk−1 π
s+1
k
ond condition transforms to v 1 =
. Note that for δ ∈ [δs , δs+1 ), the
1−δk
following condition holds: πk < πk−1 < · · · < πs+1 < 1. From Lemma 3 it then follows that
³
´s−k+2
δn
δn δαn
δn
δαn
πk−1 = αn−1
πk , πk−2 = αn−1
π
,
.
.
.
,
π
=
πk . Combining these cons+1
k
αn−1
αn−1 αn−1
ditions results in πk =
therefore is proved.
δ(1−δs )
αn k−s−1 k
(n(1−( αn−1
)
δ )−1)(1−δ)
and v 1 =
26
(n−1)(1−δs )
αn k−s−1 k
(n(1−( αn−1
)
δ )−1)(1−δ)
. The lemma
Proof of Lemma 7
Proof. As a preliminary step, note that the problem of maximising v subject to (7) and (8) is
equivalent to maximising πk subject to (7) and (8). In Lemma 3, we established that v = v 1
δv 1
δv
and πk = n−1
. Hence, πk = n−1
, and this result follows immediately.
Now, suppose β < β1 . First, consider parameter values for which γ > πk and prove by
contradiction that there is no equilibrium with price dispersion. Assume instead that there is
an equilibrium. The equilibrium must satisfy constraint k of equation (9) and hence v k ≥ γn,
where γ > πk . Then it is feasible to raise πk to the level of γ. However, by our preliminary
result, πk was already maximised, and we have a contradiction.
Next, note that if γ ≤ πk , then an equilibrium exists. This follows directly from a comparison of constraints (8) and (9). We have therefore established that an equilibrium exists if and
only if γ(β) ≤ πk (β, δ).
δv 1
,
Observe that γ(β) is independent of δ. Because v 1 is strictly increasing in δ and πk = n−1
n−1
πk (δ, β) is a strictly increasing function of δ. If δ = n , a fixed price can be supported in
equilibrium and therefore πk = 1 > γ. If δ = 0, then no collusive equilibrium exists and we
must have πk = 0 < γ. Therefore, there is a unique δ̂ such that γ(β) = πk (β, δ̂). An equilibrium
exists if and only if δ ≥ δ̂.
Proof of Lemma 8
Proof. The knot discount factors are characterized by v s = αn for s = 2, . . . , k, and equation (12). We can represent v 1 = δv 2 + 1 = · · · = δs−1 v s + δs−2 + · · · + 1 = δs−1 αn + δs−2 + · · · + 1.
Substitute v 1 into equation (12) to derive
αn − 1
δv 1 =
αn
µ
¶k−s µ
¶
n −1
αnδs−k .
n
(17)
Using Lemma 3 and substituting δv 1 from equation (17), we derive
αn − 1
πk =
αn
µ
¶k−s
αδs−k .
(18)
The s-th zone exists whenever πk ≥ γ is feasible. The value of δ which is critical value is
³ ´ 1
α k−s
δ = αn−1
. Substitute this value in equation (12) and simplify to derive
αn
γ
µ
αn − 1
αn
¶s µ ¶ k
µ
¶ µ ¶ k−i
X αn − 1 s−i α k−s n − 1
α k−s
1 s−1
+
=
.
γ
αn i =1 αn
γ
n
Define the terms
X=
αn − 1
,
αn
Y =
27
α
,
γ
(19)
and note that 0 < X < 1, Y > 1, and both X and Y are increasing functions of β. Use X and Y
to rewrite equation (19),
X sY
k
k−s
³
+ (1 − X ) X s−1 Y
k−1
k−s
+···+ X Y
k−s+1
k−s
´
=
n −1
.
n
With some manipulation, we obtain
³
Xs Y
s−1
k−s
−Y
s−2
k−s
´
³
+ X s−1 Y
s−2
k−s
−Y
s−3
k−s
´
+···+ X =
n − 1 − k−s+1
Y k−s .
n
The left hand side of this expression is increasing in β while the right hand side is decreasing in β. Also, note that if β = 0, then X = 0 which implies that the left hand side is equal
to zero. Alternatively, if β = 1, then γ = 0 and α = 1 and the left hand side becomes infinite.
Therefore, there is a unique solution β = βs . Increasing β will increase πk , decrease γ, and
make the constraint πk ≥ γ easier to satisfy (see equation (18)). Consequently, it is proved
that for β > βs the s-th zone exists. The lemma therefore is proved.
Proof of Proposition 1
Proof. The result follows directly from Lemmas 1 to 8.
Proof of Proposition 2
Proof. The result follows directly from Lemmas 1 to 8.
Proof of Proposition 3
Proof. Consider the k-period cycle σk with revenues {πs }ks=1 that solves the program (6) (9). Introduce π0 = 1 at the beginning of each cycle to form strategies σk+1 . That is, suppose
the k + 1-period cycle has revenues {πs }ks=0 where π0 = 1. We will show that the cycle σk+1
satisfies all the required constraints in (7) - (9), and that it delivers a higher value than σk .
Let the notation v s refer to values of the k-period cycle and v s0 refer to the k + 1-period
cycle. First let us show that v 10 =
1+δπ1 +···+δk πk
1−δk+1
is greater than v 1 =
π1 +δπ2 +···+δk−1 πk
.
1−δk
Define
S = π1 + δπ2 + · · · + δk−1 πk ,
then the inequity v 10 ≥ v 1 becomes
1 + δS
1 − δk+1
≥
S
1 − δk
.
(20)
Simplifying (20) leads to
S ≤ 1 + δ + · · · + δk−1 ,
which always holds because all revenues are bounded above by 1. Consequently it is proved
that v 10 ≥ v 1 .
28
Next, let us show that the proposed k + 1-period cycle satisfies all the constraints in (7) (9). Compare constraint 1 for the two cycles. With π0 = π1 = 1, we have v 1 ≥ αn and v 10 ≥ αn
for the k- and k + 1-period cycles, respectively. Because v 10 ≥ v 1 , constraint 1 of the k + 1period cycle must be satisfied. Next, let us consider the remaining constraints. Because the
right hand sides of the respective inequalities coincide, it is sufficient to prove that
0
v s+1
≥ v s ∀ s = 1, . . . , k.
This inequality ∀ s = 1, . . . , k can be presented as
(πs + δπs+1 + · · · + δk−s πk + δk−s+1 + δk−s+2 π1 + · · · + δk πs−1 )/(1 − δk+1 ) ≥
(πs + δπs+1 + · · · + δk−s πk + δk−s+1 π1 + · · · + δk−1 πs−1 )/(1 − δk ).
Define L = πs + δπs+1 + · · · + δk−s πk and R = δk−s+1 π1 + δk−s+2 π2 · · · + δk−1 πs−1 ∀ s =
1, . . . , k. The above inequality transforms to
L + δk−s+1 + δR
1 − δk+1
≥
L +R
1 − δk
.
(21)
In addition, define S 0 = R + δk L and substitute R = S 0 − δk L in equation (21) to derive:
δk−s+1 + δS 0
1 − δk+1
≥
S0
1 − δk
.
(22)
Notice that S = S 0 /δk−s+1 . Now, dividing both sides of (22) by δk−s+1 , we obtain (20). Con0
sequently, v s+1
≥ v s ∀s = 1, . . . , k, and all the constraints in (7) - (9) are satisfied. The lemma
therefore is proved.
B A consumer search model
In this section, we introduce a model of consumer search that leads to a market share relation (1) and satisfies Assumption 2.
Firms sell to a unit mass of consumers. Each consumer has a downward-sloping demand function satisfying Assumption 1. Consumers do not observe the current price vector.
Instead, their knowledge of their current price options stems from two sources. First, each
consumer has an understanding about the price distribution F and the bounds of this distribution, p and p. In the manner of Definition 1, we could impose a consistency requirement on these consumer beliefs. Second, each consumer has access to a salience relation
φ(p, p, p) : R3+ → [0, 1] that describes the probability of observing and recalling a price p
given the bounds p and p. This salience relation is described by Assumption 2’, which replaces Assumption 2 in Section 1.
Assumption 2’. The salience relation takes the following form.
(
β if p ∈ [p, p]
φ(p, p, p) =
1 if p ∉ [p, p].
where β ∈ (0, 1) is a constant.
29
(23)
Consumer search takes place in two stages. First, consumers engage in passive search.
Each consumer observes and recalls each price p j with probability φ(p j , p, p), yielding perfect information about a (possibly empty) subset of the price vector. Second, consumers
commence active search. Armed with knowledge of a vector of salient prices, each consumer
engages in costly sequential search with free recall. In each round, she decides whether to
buy at the cheapest price she has observed so far in either passive or active search, or to continue her search. Her initial search is free19 , while subsequent searches incur a cost of c per
search.
Notice that the existence of the passive search phase places the model between the benchmarks of perfect information and a standard search environment. In the perfect information setting, consumers are armed with knowledge of the full vector of available prices. Our
model collapses to this case if φ(p, p, p) = 1 ∀ p. In a typical search environment, consumers
have an understanding about the distribution of prices, without a knowledge of any specific
firm’s price. Our model reduces to this case if φ(p, p, p) = 0 ∀ p.
A strategy for consumer i is then a decision rule σci : A j → {0, 1} with σci (p j ) = 1 if consumer i chooses to buy rather than continue her search after observing a lowest price of p j
in her search so far. This costly sequential search problem is quite standard and results in a
simple consumer stopping rule. The passive search phase plays a similar role to advertising
in the Robert and Stahl (1993) model.
Under our formulation of active search, it is a weakly dominant strategy for consumers
to engage in an initial search. Suppose that consumer i has completed passive search and
recalled minimum price p j (with p j = ∞ if no prices are recalled). Suppose that she either
found no prices to be salient or decided against purchasing from a salient station. Her search
efforts so far have unearthed a minimum price of z. Her expected benefit from searching
again and then buying from the lowest price available is
¶
Z z µZ z
W (z) =
D(p j )d p j d F (x)
Z0 z x
=
D(p j ) F (p j ) d p j .
0
Her indifference point defines her reservation price, r , which depends on her beliefs, F :
Z r
W (r ) =
D(p j ) F (p j ) d p j = c.
(24)
0
Thus, each consumer i adopts the stopping rule σci (p j ) = 1 if and only if p j ≤ r . It is well
known that this stopping rule is independent of the number of remaining unsearched stations (see, for example, Stahl (1989) and Kohn and Shavell (1974)).20
19
The availability of a free initial search is not required for the results that follow. Inclusion of initial search
costs would allow us to place bounds on such search costs for the existence of the market. However, one element of our formulation is important for our results: there is no additional cost of an initial random search
relative to the cost of travel to a salient station. Otherwise, this would provide firms with an incentive to advertise high salient prices to attract custom in some circumstances.
20
Alternatively, if search is without recall it is straightforward to show that reservation prices (i) are higher
than under the free recall case and (ii) increase as the number of unsearched firms dwindles.
30
Consumer behaviour can then be summarised as follows. It is weakly dominant to perform the initial price search. Suppose that the minimum price observed by consumer i from
the combination of passive search and the initial price search yields price p j . Consumer i
then purchases from the lowest price firm (with ties broken by randomisation) if p j ≤ r and
continues her search otherwise.
This consumer search problem gives rise to market shares of the form (1) where the parameters α and γ are defined as follows
Ã
!
n−1
X
n
−
1
1
1
α(β) = β + (1 − β)L(β),
L(β) =
βn−1−k (1 − β)k
.
γ(β) = (1 − β)n ,
n
k
1+k
k=0
If firm j sets a price outside the support of F , this will surely be noticed, yielding a market
share of 0 (1) for an unusually high (low) price as per (1). If all firms set the same price, market
share is split equally. If firm j sets a price higher than her rivals, she will only attract custom
from those consumers who do not recall any prices in passive search and randomly sample
firm j . If firm j undercuts her rivals within the bounds [p, p], she will attract all consumers
who notice her either through passive search or random sampling. L(β) is the probability
that firm j is randomly sampled, conditional on not being salient. It is straightforward to
verify that α(β) and γ(β) satisfy Assumption 2.
C Profits and punishment
An important implication of our model is that if consumers are imperfectly attentive, it is
possible that price dispersion could make collusion easier to sustain. The mechanism is
quite intuitive. Price variation makes it harder for consumers to detect modest price undercutting. This affects the sustainability of collusion by reducing the payoffs to firms deviating
from cartel policies. However, if price dispersion is effective at obfuscating the price process,
it is also possible that the ability of the cartel to punish a defector is impacted. In this section,
we examine the trade-off involved if this is the case.
First, we introduce some additional notation and definitions.21 Let p ≡ (p 1 , p 2 , . . . , p n )
denote a price vector, and p j the associated vector with p j removed. Let ~
p ≡ (p, p, . . . , p)
describe a vector of n prices in which each firm chooses the same price p. Let ~
p j be the
corresponding n − 1 price vector with the price of firm j omitted. Profits for firm j are then
given by π j (p; p, p) = R(p j )s j (p; p, p). Define π∗j (p j ; p, p) ≡ supp 0 π j (p 0 , p j ; p, p). Letting p̆ j
minimise π∗j (p j ; p, p), minmax profits for j are then given by π∗j (p̆ j ; p, p).
We adopt the following restrictions on the firm’s profit function.
Assumption 3. For all j and k 6= j ,
∂s j (p; p, p)
∂p k
21
Our exposition is closely related to Lambson (1987).
31
≥ 0.
Assumption 4. For all j and all p,
s j (~
p, p, p) =
1
.
n
Assumption 5. There exists p̃ ≤ p such that for all p ∈ [p̃, p] and all j ,
π j (~
p; p, p) > π∗j (~0 j ; p, p).
Assumption 6. For all j and all p < p̂,
π∗j (~
p j ; p, p) − π j (~
p; p, p)
is nondecreasing in p.
Together, Assumptions 3 - 6 place milder restrictions on the profit function than Assumption 2 in the body of the paper. Assumption 3 imposes monotonicity and implies that
marginal cost pricing by all other firms minmaxes firm j ’s profits. By Assumption 4, the market is shared equally if all firms set the same price. Assumption 5 ensures that it is possible
for firms to earn collusive profits that are greater than the minmax profit level. Assumption
6 suggests that the returns from deviation are weakly increasing in prices.
A punishment for firm j is a sequence of price vectors τ j = {p(t , j )}∞
t =0 , and a simple penal
code is a vector of punishments τ = (τ1 , τ2 , . . . , τn ). The value to firm j if the punishment for
P
t
firm i is followed is V j (τi ; p, p) = ∞
t =0 δ π j (p(t , i ); p, p). A simple penal code is credible if
∀i , j , and t 0 ,
∞
X
t =0
δt π j (p(t 0 + t , i ); p, p) ≥ π∗j (p j (t 0 , i ); p, p) + δV j (τ j ; p, p).
(25)
If Firm j receives minmax profits in every period, she obtains her security value v(p, p) ≡
π∗j (p̆ j ; p, p)/(1 − δ). For any credible penal code τ, and for all j , we must have V j (τ j ; p, p) ≥
v(p, p). Define a security level punishment τ j as the punishment that achieves the value
V j (τ j ; p, p) = v(p, p), and the associated simple penal code as τ. An optimal penal code is a
simple penal code that minimises V j (τ j , p, p), subject to the constraint that it is credible.
Consider the infinitely repeated two period cycle {p a , p b } where p a , p b ∈ [p, p] and p ≥
p̃.22 A fixed price is a special case in which p a = p b . Let Φ(p, p) be the set of two-period
cycles sustainable by τ. Then, {p a , p b } ∈ Φ(p, p) if and only if
π j (~
pa ; p, p) + δπ j (~
pb ; p, p)
1+δ
π j (~
pb ; p, p) + δπ j (~
pa ; p, p)
1+δ
22
≥ (1 − δ)π∗j (~
pa, j ; p, p) + δπ∗j (~0 j ; p, p),
(26)
≥ (1 − δ)π∗j (~
pb, j ; p, p) + δπ∗j (~0 j ; p, p).
(27)
The extension to a finite k period cycle is immediate.
32
Proposition 4. Suppose Assumptions 1 and 3-6 hold. Fix p a > p̃ such that {p a , p a } ∈ Φ(p a , p a )
and let δ∗ (p a ) be the solution to
π j (~
pa ; p a , p a ) = (1 − δ∗ (p a ))π∗j (~
pa, j ; p a , p a ) + δ∗ (p a )π∗j (~0 j ; p a , p a ).
(28)
Then there exists a price p b < p a and a discount factor δ < δ∗ (p a ) such that {p a , p b } ∈ Φ(p b , p a )
if equations (29) and (30) hold:
¢
δ∗ (p a ) ¡
π j (~
pb ; p b , p a ) − π j (~
pa ; p a , p a ) >
(29)
∗
1 + δ (p a )
³
´
³
´
(1 − δ∗ (p a )) π∗j (~
pa, j ; p b , p a ) − π∗j (~
pa, j ; p a , p a ) + δ∗ (p a ) π∗j (~0 j ; p b , p a ) − π∗j (~0 j ; p a , p a ) ,
¡
¢
1
π
(~
p
;
p
,
p
)
−
π
(~
p
;
p
,
p
)
>
(30)
j
a
j
a
a
a
b
b
1 + δ∗ (p a )
³
´
³
´
(1 − δ∗ (p a )) π∗j (~
pb, j ; p b , p a ) − π∗j (~
pa, j ; p a , p a ) + δ∗ (p a ) π∗j (~0 j ; p b , p a ) − π∗j (~0 j ; p a , p a ) .
Proof. 1. By Assumption 3, security level payoffs are given by
v(p, p) =
π∗j (~0 j ; p, p)
1−δ
.
(31)
2. Let δ∗ (p a , p b ; p, p) be the lowest discount factor satisfying (26) and (27). Notice that the
left hand side of (27) is increasing in δ and the right hand side of (27) is decreasing in δ.
Therefore, (27) is satisfied for δ ≥ δ∗ (p a , p b ; p, p).
Differentiating (26) by δ, the following condition ensures that (26) will also be satisfied
for δ ≥ δ∗ (p a , p b ; p, p):
π j (~
pb ; p, p) − π j (~
pa ; p, p)
(1 + δ)2
pa, j ; p, p).
> π∗j (~0 j ; p, p) − π∗j (~
(32)
We can always choose p b sufficiently close to p a to satisfy (32). Combining these results,
there exists p b < p a such that (26) and (27) are satisfied for δ ≥ δ∗ (p a , p b ; p, p).
3. Next, we construct an optimal penal code τ. To do so, first define the price p u and integer
u such that
u−2
X
δt π j (~0; p, p) + δu−1 π j (~
pu ; p, p) + δu
t =0
pb ; p, p)
π j (~
pa ; p, p) + δπ j (~
1 − δ2
= v(p, p).
(33)
There are two possibilities for the relationship between minmax profits and profits in the
initial, harshest, phase of punishment. If π j (~0; p, p) = π∗j (~0 j ; p, p), then there is a Nash equilibrium to the stage game that minmaxes each player. In this case, we use infinite repetition
of this stage game Nash equilibrium as punishment by setting u = ∞. Note that this was the
situation we examined in the body of the paper.
33
Alternatively, if π j (~0; p, p) < π∗j (~0 j ; p, p), then we must show that there exists an integer
u and a price p u < p b that satisfies (33). Initially suppose p b = p a . Next, note that Assumption 5 implies that π j (~
pb ; p, p) > π∗j (~0 j ; p, p). It follows that there must exist a pair u and
p u < p a that satisfies (33) in this case. Now, using the continuity of R(p) from Assumption 1,
there exists a p b < p a such that this condition is also satisfied.
For {p a , p b } ∈ Φ(p, p) and each j , then define τ j as follows
p(t , i ) =


~

0
0 ≤ t ≤ u −2
~u
p
t = u −1


{~
pa ,~
pb } t ≥ u.
Note that we must have u ≥ 1. Otherwise, the penal code consists only of the infinite price
cycle {p a , p b } which has a value greater than the security value by Assumption 5. In points 4.
and 5., we consider the case where u ≥ 2. In 6., we consider u = 1.
4. Let us show credibility of the optimal penal code τ for the case u ≥ 2 by establishing (25).
In period t 0 ≤ u − 2, we can rewrite (25) as
u−2−t
X 0
0
pu ; p, p) + δu−t
δt π j (~0; p, p) + δu−1−t π j (~
0
π j (~
pa ; p, p) + δπ j (~
pb ; p, p)
1 − δ2
t =0
(34)
≥ π∗j (~0 j ; p, p) + δv(p, p)
For t 0 = 0, by (33), equation (34) is satisfied with equality. For 0 < t 0 ≤ u−2, because π j (~0; p, p) <
min{π j (~
pu ; p, p), π j (~
pa ; p, p), π j (~
pb ; p, p)}, (34) must also be satisfied.
0
For t ≥ u, inequality (25) follows if {p a , p b } ∈ Φ(p b , p a ). We establish the conditions for
this result in step 5, below.
Finally, for t 0 = u − 1, condition (25) requires
π j (~
pu ; p, p) + δ
π j (~
pa ; p, p) + δπ j (~
pb ; p, p)
1 − δ2
≥ π∗j (~
puj ; p, p) + δv(p, p)
(35)
≥ π∗j (~
pb, j ; p, p) + δv(p, p)
(36)
Notice that if {p a , p b } ∈ Φ(p b , p a ), then from (27)
π j (~
pb ; p, p) + δ
π j (~
pa ; p, p) + δπ j (~
pb ; p, p)
1 − δ2
Because p u > p b , Assumption 6 and (36) imply (35).
5. Now, for u ≥ 2, we compare the sustainability of a fixed price path with that of a cycle.
Consider first a candidate fixed price equilibrium with price p a . In the candidate equilibrium, p = p = p a . Consequently p a is sustainable if and only if
π j (~
pa ; p a , p a )
1−δ
≥ π∗j (~
pa, j ; p a , p a ) +
δ
π∗ (~0 j ; p a , p a ).
1−δ j
Let δ∗ (p a ) solve (37) with equality and rearrange to obtain (28).
34
(37)
Next consider a candidate price cycle equilibrium with prices p a and p b for some p b <
p a . In the candidate equilibrium, p = p b and p = p a . The cycle {p a , p b } is sustainable if and
only if
π j (~
pa ; p b , p a ) + δπ j (~
pb ; p b , p a )
1+δ
π j (~
pb ; p b , p a ) + δπ j (~
pa ; p b , p a )
1+δ
≥ (1 − δ)π∗j (~
pa, j ; p b , p a ) + δπ∗j (~0 j ; p b , p a ),
(38)
≥ (1 − δ)π∗j (~
pb, j ; p b , p a ) + δπ∗j (~0 j ; p b , p a ).
(39)
The candidate price cycle equilibrium is then sustainable for a greater range of discount
factors if the inequalities in (38) and (39) are strict for δ = δ∗ (p a ). Combining (28), (38) and
(39), and applying Assumption 4 we obtain the conditions (29) and (30).
6. Finally, consider the case u = 1. Credibility at time t = 0 is established by the same argument examined for t 0 = u − 1 in point 4. Credibility at time t ≥ 1 and sustainability are both
demonstrated by repeating the argument of point 5.
Proposition 4 establishes the conditions under which collusion is supportable for a greater
range of discount factors using a price cycle rather than a fixed price. Equations (29) and (30)
highlight the trade-off between the profitability of deviation and punishment across a cycle
and a fixed price. The left hand side of each equation measures the difference between the
average value of collusion with a cycle and with a fixed price. Because the cycle involves
spending some time setting the lower price p b , average profitability under the cycle path
will be lower. However, this difference will be (i) small if p b is not too far below p a ; and (ii)
less important if we consider cycles of greater length.
The first term on the right hand side is the difference between deviation profits under a
cycle and a fixed price. Our argument in the body of the paper is that the profits from deviation may be markedly higher in the context of a fixed price path relative to a price cycle
because price dispersion makes it difficult for consumers to distinguish small price differences that lie within their realm of experience. By this argument, we may expect this term to
be negative and substantial.
The second term on the right hand side is the difference between punishment profits
under the cycle and the fixed price path. In our specification of Section 2, a zero profit stage
game equilibrium was available and this term dropped out. Here, we allow punishment payoffs to depend on the extent of price dispersion. Intuitively though, the influence of price
dispersion is likely to be small. To illustrate, suppose firms have enjoyed a spell of collusion.
One firm then deviates, triggering a price war in which all firms set price equal to marginal
cost for a sustained period. This price war represents a sharp break in the pricing pattern of
the market and is likely to attract the attention of consumers regardless of the precise form of
collusive pricing. If a single firm were to depart from the proscribed punishment by raising
price, it is likely that consumers will notice that this firm offers a price that is above the price
of its rivals. This argument pertains whether collusion had taken the form of a fixed price or
a price cycle.
35
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