Collusion with costly consumer search∗
Vaiva Petrikaitė†
2014, May (Revised version)
Abstract
The effect of market transparency on the stability of collusion depends on whether a market
is opaque on the demand or the supply side. The economic literature suggests that an increase
in market transparency on the supply side facilitates collusion. On the contrary, in this paper,
I show that a cartel is more stable if a market is opaquer on the demand side. To model
market transparency, I assume that a consumer pays a positive search cost per firm to learn
the features of horizontally differentiated products. Because of the search cost, on average, a
consumer compares less alternatives. Thus, an increase in the search cost decreases market
transparency on the demand side. Both the gain from the deviating form collusion and the
punishment that follows the deviation decrease with market opacity. However, the deviation
gain is more sensitive to market opacity than the punishment. Consequently, cartel stability
decreases when market transparency decreases.
Keywords: Sequential search, non-sequential search, cartel, collusion, search costs, horizontal differentiation
JEL classification: D43, L13, L41
∗
Financial support from the Marie Curie Excellence Grant MEXT-CT-2006-042471 is gratefully acknowledged. I
am grateful to José L. Moraga-González, Marco Haan, Régis Renault, Maarten Janssen, Jan Boone, Bert Schoonbeek,
Peter van Santen, Shu Yu, Wim Siekman, Tadas Bružikas the participants of EARIE 2012, the seminar participants
in IESE Business School, the editor and a referee for helpful comments and suggestions.
†
Faculty of Economics and Business, University of Groningen, Nettelbosje 2, 9747AE Groningen, the Netherlands;
e-mail: [email protected]
1
1
Introduction
The economics literature suggests that collusion is harder or even infeasible to sustain if a market is
opaque from the point of view of cartel members.1 If the sellers cannot observe each others actions
perfectly, then the punishment that follows a deviation from collusion becomes softer, which makes
deviating more attractive. Meanwhile, an answer to the question how market transparency from the
consumer point of view affects cartel stability is not that trivial. This happens because both the gain
from deviating and the punishment that follows the deviation are affected by market transparency.
A market is said to be opaque on the demand side if a consumer encounters difficulties to observe
and compare existing alternatives. In such a market, firms have more market power and earn higher
profits in a competitive equilibrium. Thus, the punishment that follows a deviation is softer, which
makes deviating more likely. In addition, in an opaque market, fewer consumers observe a deviation
from collusion. Hence, the gain from deviating and the incentives to deviate are lower. Because
both the punishment for deviating and the gain from deviating decrease, it is not obvious whether
cartel stability increases or decreases with market opacity. If the deviation gain decreases more than
the punishment, then collusion becomes more sustainable if market transparency decreases. On the
contrary, if the deviation gain decreases less than the punishment, then a cartel is less stable if
market opacity increases.
To the best of my knowledge, there are only two papers (Nilson, 1999 and Schultz, 2005) where
it is studied how cartel stability varies with market transparency on the demand side. The findings
in the papers indicate that the degree of product differentiation and the method that is used to
model market transparency have a very strong impact on the sensitivity of the deviation gain and
the punishment to market transparency. Nilson (1999) studies a duopoly market by using the model
of Burdett and Judd (1983) for homogenous products. In his set-up, a fraction of consumers observe
the prices of both firms, whereas the second fraction of consumers have to pay a positive search
cost per firm to learn the offers of the sellers. Because of costly search, a consumer compares less
alternatives on average. Therefore, it is said that market transparency decreases from the consumer
point of view when the search cost increases. Nilson (1999) shows that cartel stability decreases
when market transparency decreases. On the contrary, Schultz (2005) finds that in a horizontally
differentiated product market a cartel is more stable if the market becomes more opaque. In his
set-up, consumers are uniformly distributed on a Hotelling line and the firms are located at the
ends of the line. The degree of market transparency is modeled by assuming that a fraction of
consumers do not observe actual prices and for exogenous reasons can visit only one seller. Thus,
market transparency decreases when the share of the uninformed consumers goes up.
Schultz (2005) uses a very specific method to model horizontal product differentiation. In his
model, a valuation (match value) of a product equals the difference between a constant and the total
transportation costs. Hence, if a consumer is close to one firm, which implies a high valuation for
the product of the seller, then the valuation of the other product is low. Thus, the valuations of the
1
Stigler (1964); Green and Porter (1984); Abreu et al. (1985); Kandori and Matsushima (1998); Compte (2002)
2
products are negatively correlated in the model, which gives the most extreme horizontal product
differentiation.
In this paper, I also analyse how cartel stability varies with market transparency in a horizontally
differentiated product market. However, I employ a different method to model product differentiation. Namely, I assume that consumers’ valuations of products are independently and identically
distributed among consumers and products like in Perloff and Salop (1985). Thus, a low valuation of
one product does not imply that the valuation of another product is low. To model market opacity,
I assume that a consumer has to pay a positive search cost per firm to learn both her valuation of
a product and the price. Then, similarly to Nilson (1999), market transparency decreases when the
search cost goes up. In the model, a consumer searches products sequentially with perfect recall
like in Wolinsky (1986) and Anderson and Renault (1999) or samples the varieties non-sequentially
similarly to Anderson et al. (1992) and Moraga-González et al. (2013). The number of products that
a consumer samples is determined by optimal sampling and stopping rules. Therefore, contrarily to
the framework of Schultz (2005), the consideration set of a consumer is defined endogenously. I find
that the deviation gain is more sensitive to the market transparency than the deviation loss, and
the critical discount factor above which collusion is sustainable decreases with the search cost.
To account for the effect of the method that is used to model market transparency, I study a
duopoly model with randomly distributed match values (section 6). In the set-up, a fraction of
consumers observe their the match values for products but cannot observe actual prices and they
can visit only one seller. The rest of consumers are fully informed. Then, similarly to Schultz (2005),
market transparency decreases with the share of the uniformed consumers. Contrarily to Schultz
(2005), I find that the deviation gain decreases less with market opacity than the punishment, and,
therefore, cartel stability decreases with market opacity.
The difference between the findings in section 6 and the main result of Schultz (2005) stems from
a difference in demand price elasticities. In the random utility model, the demand of a firm is more
elastic to its own price than in the Hotelling model. Because of this, the effect of market transparency
on the competitive profit, and consequently on the punishment is higher in the random utility model
than in the Hotelling model. Hence, when match values of products are distributed independently
and market transparency decreases, the punishment decreases faster than the deviation gain, which
leads to a decreasing cartel stability.
The rest of the paper is organized as follows. In the subsequent section, I describe the main
modeling assumptions. Competitive and joint profit maximizing prices and profits are derived in
section 3. How the stability of a cartel changes when a search cost varies is analysed in sections 4
and 5. In section 6, I show that the effect of market transparency on cartel stability can be different
when market transparency is determined by the fraction of uninformed consumers. Finally, some
concluding remarks are written in section 7. Longer proofs and derivations are delegated to the
appendices of the paper.
3
2
Model description
On the demand side there is a unit mass of consumers. A consumer wants to buy one unit of a
product and she can choose from n ≥ 2 horizontally differentiated varieties. Every variety is sold
by a distinct seller. All the firms have the same constant unit production cost that is normalized to
zero. Consumer i who buys product j gets utility uij :
uij = εij − pj
(1)
where εij is the match value by consumer i of product j and pj stands for the price of the product.
Consumers differ in their tastes and it is common knowledge that match values are distributed
identically an independently across consumers, varieties and time according to a uniform distribution
in the interval between zero and one.2
A consumer prefers to buy the alternative that gives her the highest utility. However, she does
not know what variety and at what price is sold in each shop exactly. The customer has expectations
about prices and has to incur a positive search cost s per shop to learn exact utilities. In this paper,
I analyse two consumer search frameworks: sequential and non-sequential search.
Sequential search. In the sequential search framework, consumers visit firms with perfect recall
and can terminate their search after sampling any number of firms. Suppose that consumers expect
that the price of all n varieties is the same and equals p∗ . Then the distribution of utilities of the
products is identical. Hence, by the optimal stopping rule (see Kohn and Shavell (1974)), consumers
use the reservation utility of a firm to decide when to terminate their search. The reservation utility
of a seller equals x̄ − p∗ , where x̄ is computed by using equation (2).
ˆ
1
(ε − x̄) dε = s
(2)
x̄
The left hand-side (LHS) of (2) shows the expected gain from search when the highest observed
match value equals x̄, and the search cost is on the right hand-side (RHS) of the equation. Then if
the maximum observed utility is greater than x̄−p∗ , then a consumer terminates her search and buys
the product that provides her with the highest observed utility. Otherwise, the consumer continues
searching further. If the consumer samples all n firms and the highest observed utility is negative,
then the customer does not buy anything.
2
The assumption that purchase decisions are independent from the choices in the past is realistic in markets
where the assortment in shops changes between the purchases of a consumer. For instance, the producers of home
appliances, computers, phones and cars update their models due to rapid technological progress. Thus, a buyer
always finds different offers whenever she comes to buy a washing machine, a fridge or a digital camera. Past
purchase decisions and experience may imply that a consumer has higher (or lower) preferences over some brands.
However, due to new models brand loyalty affects only the search order of a consumer but not the value of a particular
product. Because purchase histories of consumers differ, the search orders across consumers vary too.
4
Non-sequential search. In a non-sequential search model, a consumer samples 2 ≤ k ≤ n firms
before deciding what and whether to buy. Suppose that the expected price of every variety equals
p∗ . Then the consumer expects to buy one of the k varieties if max {εi − p∗ }i=1,...,k ≥ 0; otherwise
she will not buying anything. The net benefit of the consumer after sampling k varieties is
ˆ
1
(ε − p∗ ) kεk−1 dε − ks
(3)
p∗
For any fixed p∗ the expression (3) is concave in k.3 Therefore, there is an optimal number of firms
that a consumer samples, and this number is defined by equation (4).4
ˆ
1
(ε − p∗ ) εk−2 (kε − k + 1) dε = s
(4)
p∗
where the LHS of (4) is the gain from sampling one more firm, and the search cost of one firm is on
the RHS of the equation.
Additionally, I assume that s is sufficiently low such that:
• in the sequential search model, a consumer samples at least one firm,
• in the non-sequential search set-up, a consumer always samples at least two firms.
Because the distribution of match values is identical across the varieties, and all the firms have
the same production costs, in the subsequent sections, I study symmetric competitive and collusive
market equilibria where all the firms charge the same prices.
3
One period interaction
3.1
Non-cooperative equilibrium
Let p∗ be the expected price that all the firms charge in the absence of collusion. To distinguish
among the two different search regimes all the equilibrium prices and profits that are related to
sequential search have ∼ sign and all the equilibrium prices and profits in the non-sequential search
framework have ∧ above.
Sequential search. In the sequential search model, the competitive symmetric equilibrium is like
the one in Wolinsky (1986). Here I provide the summary of it. Consider a firm j that deviates to a
price p 6= pe∗ and the rest of shops charge pe∗ . Firm j may be visited in the first, second and in any
other position up to the nth with the probability 1/n. A consumer who has reached firm j observes
3
See Appendix A for more details.
In the subsequent analysis, if the value of k that satisfies (4) is not an integer number, then I take the largest
integer that does not exceed the solution to (4).
4
5
the deviation and interprets p as the deviation price of firm j only. Therefore, she expects to see pe∗
in other shops. As a result, the consumer terminates her search at firm j if εj − p is greater than or
equal to x̄ − pe∗ . Otherwise, the customer goes to another firm. If the consumer has arrived at firm
j, then the utilities at preceding shops must have been less than x̄ − pe∗ (or max {εi }i<j < x̄). The
probability that a single consumer has arrived at firm j and terminated her search there equals the
“fresh demand” of firm j. I label this demand fj .
n
1 X i−1
1 1 − x̄n
fj =
(1 − x̄ + pe∗ − p)
x̄ (1 − (x̄ − pe∗ + p)) =
n i=1
n 1 − x̄
The consumers who continue searching after visiting firm j may still buy from firm j. This
happens if the buyers visit all the sellers and find that the utility at firm j is the highest and is
greater than zero. The number of consumers who return to firm j and buy there are called the
“returning demand” of firm j and is labeled rj .
h
i ˆ
∗
∗
rj = Pr max {εl − pe , 0}∀l6=j ≤ εj − p < x̄ − pe =
p
x̄−e
p∗ +p
(ε − p + pe∗ )n−1 dε
The profit function of firm j equals the total income of firm j. I label this function πj (p).
πj (p) = p (fj + rj )
(5)
In equilibrium, firm j sets p = pe∗5 and the first-order condition of firm j simplifies to equation
(6).
1 − x̄n ∗
pe − (e
p∗ )n = 0
(6)
1 − x̄
After some arithmetic manipulations one can get that the equilibrium profit of the firm π
e∗ is
1−
pe∗
π
e =
(1 − (e
p∗ ) n )
n
∗
(7)
Non-sequential search. In the non-sequential search framework, a consumer samples firms randomly and firm j can be sampled among k firms with probability k/n. Suppose that n − 1 firms
charge a price pb∗ and firm j sets p < pb∗ . A consumer who samples the seller buys from firm j if
εj − p ≥ max {εi − pb∗ , 0}j=1,...,k−1 , which happens with probability
ˆ
ˆ
1
∗ k−1
dε +
1−b
p∗ +p
1−b
p∗ +p
p
(ε − p + pb )
1
∗ k
dε = pb − p +
1 − (b
p) .
k
∗
The pay-off of the deviant seller equals
5
See Moraga-González and Petrikaitė (2013) footnote 12 for the discussion about the quasi-concavity of the pay-off
function.
6
kp
πj =
n
1
∗ k
pb − p +
1 − (b
p)
k
∗
(8)
The pay-off function (8) is strictly concave in p (∂ 2 πj /∂p2 = −2k/n < 0) and the unique
equilibrium price pb∗ is defined by equation (9).
1 − pb∗ k − (b
p∗ )k = 0
(9)
After setting p = pb∗ the competitive equilibrium pay-off of the firm becomes
pb∗ ∗ k
1 − (b
p)
π
b =
n
∗
3.2
(10)
Joint price setting
No matter the search behaviour of consumers, the pay-offs that have been derived in section 3.1
increase if the symmetric equilibrium prices go up. Horizontally differentiated products selling
firms exert negative pricing externalities on each other. Hence, profits would increase if the firms
maximized their joint profit instead of competing with each other. In this section, I derive the joint
profit-maximizing prices and the pay-offs of colluding sellers. All the prices and profits that are
related to collusion have a superscript c.
Sequential search. In a collusive set-up, sequentially searching consumers expect that all the
firms charge the same price pec and sample the sellers randomly. The buyers do not know for sure if
the firms coordinate their pricing decisions. Therefore, as before, any deviation price they observe
is attributed to an individual firm and the expectations about the prices of not-yet-visited shops
are not updated. Meanwhile, the cartel members look for an optimal vector of prices {p1 , ..., pn } 6=
{e
pc , ..., pec }.
As all the firms set the same price pec in the symmetric collusive equilibrium, for the derivation
of the first order condition it is sufficient to derive the joint pay-off function by assuming that all
the firms but firm j set pec and firm j sets a price p 6= pec .6 By using the same reasoning as in section
3.1, I obtain that the pay-off function of firm j is
πj = p
1 1 − x̄n
(1 − (x̄ − pec + p)) +
n 1 − x̄
ˆ
x̄−e
pc
c n−1
0
(ε + pe )
dε
(11)
If firm j colludes then it takes into account the effect of p on the profits of other firms. Let
us take shop h 6= j. This shop charges pec . Firm h can be sampled the first, second up to the nth
with probability 1/n. Provided that firm h is sampled the lth , the probability that firm j has been
6
In Appendix B, I show that a symmetric collusive price gives a local maximum of the profit of a cartel if there
are two (in case of non-sequential search - three) firms in the market. Numerical simulation results confirm that the
same holds for a higher number of firms and that the symmetric equilibrium price gives a global maximum of the
joint pay-off.
7
sampled before firm h is (l − 1) / (n − 1). With probability (n − l) / (n − 1) firm j has not been
sampled before firm h. By considering that a consumer terminates her search at firm h if εh ≥ x̄
and that firm h may be sampled in any position from 1 to n, I obtain that the fresh demand of firm
h is
n−1
n
X
X
n−l
l−1
c
l−2
(x̄ − pe + p) x̄ (1 − x̄) +
x̄l−1 (1 − x̄)
fh =
n
(n
−
1)
n
(n
−
1)
l=1
l=1
Firm h also sells to consumers who do not terminate their search at it but eventually buy there after
searching all the sellers in the market. The probability that these consumers buy from the firm gives
the returning demand of firm h, which is
h
i
rh = Pr max {εi − pec , εj − p, 0}i={1,2,...,n}\{j,h} < εh − pec < x̄ − pec
ˆ x̄−epc
(ε + pec )n−2 (ε + p) dε
=
0
The joint pay-off of the cartel is the sum of the pay-offs of all n firms:
Π = πj + pec (n − 1) (fh + rh )
In equilibrium, firm j sets p = pec and the first-order condition of the cartel simplifies to (12).
∂Π = 1 − (e
pc )n (n + 1) = 0
n
∂p p=epc
(12)
which gives the collusive price pec = (n + 1)−1/n . The profit of one coalition partner equals the ratio
between the joint profit and n. This profit is denoted by π
ec .
π
ec =
ec
Π
pec
= (1 − (e
pc )n ) = (e
pc )n+1
n
n
(13)
Non-sequential search. When consumers search firms non-sequentially and firms collude, the
customers expect that all the sellers charge the same collusive price pbc . To derive this price, I again
consider firm j that deviates to price p < pbc when all other sellers charge pbc . A consumer samples
firm j with probability k/n. Thus, by following the arguments in the preceding section, the profit
of the deviating firm equals
kp c
1
c k
πj =
pb − p +
1 − (b
p)
(14)
n
k
The profit of firm h 6= j is also affected by the deviation price of firm j. With probability
k (k − 1) / (n (n − 1)) a consumer samples both firm h and firm j. With probability k (n − k) / (n (n − 1))
a consumer samples firm h but does not sample firm j. If firm j is not sampled, then firm h sells
to the consumer if εh − pbc > max {εi − pbc , 0}i=1,...k−1 . If firm j is sampled with firm h, then firm h
sells to the consumer if εh − pbc > max {εi − pbc , εj − p, 0}i=1,...k−2 . As a result, the pay-off of firm h
8
equals
ˆ 1−bpc
ˆ 1−bpc
(k − 1)
c k−2
c k (n − k)
(ε + pb ) (ε + p) dε + pb
(ε + pbc )k−1 dε
πh = pb
n (n − 1)
n
(n
−
1)
0
0
pbc
=
(n − 1) 1 − (b
pc )k + k (p − pbc ) 1 − (b
pc )k−1
n (n − 1)
ck
and the joint pay-off of the cartel is
Πc = πj + (n − 1) πh
The collusive price pbc is defined by the first-order condition (15).
1 − (k + 1) (b
pc ) k = 0
(15)
Hence, the collusive price pbc equals (k + 1)−1/k and the profit of one cartel member is
k
bc
pbc Π
c k
=
1 − (b
p ) = (b
pc )k+1
π
b =
n
n
n
c
4
(16)
Collusion
The individual profit maximizing price differs from the joint profit maximizing price, provided that
other firms charge pc . Hence, a seller has incentives to deviate from the collusive equilibrium and
a cartel does not get established if the game lasts only one period. However, deviating is not
necessarily profitable from a long term perspective. If all sellers interact in the market for several
periods, then the deviant may be punished by other firms in the periods after deviation. If the
punishment is sufficiently hard, then the seller gives up its plans about deviating. In this section,
I assume that firms interact for an infinite number of periods and apply the standard grim trigger
strategy. According to this strategy, the firms set the joint profit maximizing price pc every period
if all coalition members do this in the preceding periods. If at least one shop deviates from the
collusive price, then the sellers revert to the one period Nash equilibrium price for the rest of the
time.
The expectations of consumers about the prices affect the number of sampled sellers in the nonsequential search model and the decisions on search termination if consumers search sequentially.
The consumers who visit a deviant in a deviation period observe a deviation price pd and know
that the next period is a punishment period when all the firms set p∗ . The consumers who do not
observe the deviation are not aware that the firms have turned to punishment before searching the
first seller. However, after an uninformed consumer samples the first firm, she sees that the seller
charges p∗ . Then the consumer realizes that the cartel has collapsed and updates her expectations
9
about the prices of other firms.7
A single unilateral deviation does not affect the chosen prices of other firms and expectations
of consumers about the prices of not-yet-visited firms. Thus, the buyers expect that other sellers
charge pc if they see pd in one shop. As a consequence, for the derivation of the optimal deviation
prices ped and pbd I use the pay-off functions (11) and (14), where the price p is replaced with ped
and pbd respectively. Then the profit maximizing deviation prices are defined by the corresponding
first-order conditions.8
By deviating from collusion, a firm gets a deviation gain π d − π c . However, after both sellers turn
to the punishment, the profit of the firm decreases by π c −π ∗ for every punishment period. Therefore,
P
t
c
∗
by deviating the firm knows that it will experience a deviation loss that equals ∞
t=1 δ (π − π ),
where δ is a discount factor. I define δ ∗ that makes the deviation gain equal to the deviation loss.
−1
This discount factor equals δ ∗ = π d − π c π d − π ∗ . If an actual discount factor is above this
threshold value then a cartel is sustainable. Otherwise, the firms never start colluding.
Proposition 1. Suppose that colluding firms apply the standard grim trigger strategy, and
• consumers search sequentially. Then the critical discount factor above which collusion is
sustainable decreases with the search cost.
• consumers search non-sequentially, and the search cost is such that in a competitive equilibrium a consumer samples all the firms. Then the critical discount factor above which collusion is
sustainable decreases with the search cost.
When consumers search sequentially, the collusive profit does not depend on the search cost
while the competitive profit increases with s and the deviation profit decreases with s. Therefore,
both the deviation gain and the one-period deviation loss (e
πc − π
e∗ ) decrease with the search cost.
Nevertheless, either the deviation gain is substantially smaller than the one-period deviation loss or
(and) the competitive profit increases with s less than the deviation profit decreases.9 Therefore,
the critical discount factor δe∗ decreases with s.
When consumers search non-sequentially, the number of firms that a consumer samples depends
on the search cost and the expected prices. If the firms set the joint profit-maximizing price, then
for the same search cost consumers sample less firms compared to the competitive equilibrium. In
proposition 1, I study the case when the search cost is sufficiently small and a consumer samples
all the sellers when collusion is absent. Both the deviation gain and the one-period deviation loss
decrease with the search cost. However, the deviation gain is more sensitive to the search cost than
the deviation loss. Therefore, δb∗ decreases with s. Simulation results suggest that the same result
holds for other ranges of s where consumers do not sample all the sellers in a competitive equilibrium
(Figure 1).
7
Another option is to assume that all the prices are observed in the end of every period by everyone.
The exact expressions of both deviation prices are given in the proof of proposition 1.
e∗
δ
πd
π∗
9
The sign of ∂∂s
is the opposite to the sign of ∂e
πc − π
e∗ ) + ∂e
ed − π
ec .
∂x (e
∂x π
8
10
Figure 1: The values of δb∗ for different s when the number of firms decreases with s both in the
collusive and in the competitive equilibrium.
5
Stick-and-carrot strategy
The grim trigger strategy is not the only possible strategy that makes a cartel stable. The set of
possible punishment and reward strategies depends on market conditions such as an actual discount
factor, the number of firms, etc. For instance, Abreu (1986) shows that if, in a Cournot market, the
discount factor is too low to assure cartel stability with the grim trigger strategy, then the cartel
can be sustainable if firms apply a two period stick-and-carrot strategy. According to the stickand-carrot strategy the sellers set very low prices or high quantities in the punishment period and
later return to a collusive equilibrium. The punishment prices and quantities are chosen in the way
that they also prevent deviations from the punishment. In this section, I check how cartel stability
changes with the search cost if sellers apply a prolonged stick-and-carrot strategy with non-negative
prices. To construct this strategy, I follow Häckner (1996) and assume that after a deviation from
collusion firms set prices equal to their marginal costs for τ ≥ 1 periods. If all the sellers comply
with the punishment, then in period τ + 1 the collusive equilibrium is established again. If there are
deviations in the punishment stage, then the punishment starts all over again.
Under the prolonged stick-and-carrot strategy, a firm does not deviate from collusion if the gain
from one period deviation is lower than the total deviation loss. In other words, the deviation from
a cartel set-up does not happen if inequality (17) is satisfied.
1 − δτ
δ τ +1 c
δ (1 − δ τ ) c
δ
c
π − δ
·0+
π =
π
π −π ≤
1−δ
1−δ
1−δ
1−δ
d
c
(17)
If all the sellers set their prices equal to their marginal costs, then a firm can set a higher price
and earn a positive profit because of horizontal product differentiation. Therefore, the punishment
must be such that it prevents the deviations from the punishment too. If the deviation from the
punishment happens, then it happens in the first punishment period. Because after any defection
the counting of punishment periods starts all over again, no deviation from the punishment happens
11
if inequality (18) is satisfied.
π
d0
1 − δτ
δτ c
δ τ +1 c
1 − δ τ −1
·0+
π − δ
·0+
π = δτ πc
≤ δ
1−δ
1−δ
1−δ
1−δ
(18)
where π d0 is the profit of a deviating firm if it deviates to price pd0 in the punishment phase.
The critical discount factor that satisfies (17) and (18) with equalities equals
δ0 =
πd − πc
.
π d − π d0
Proposition 2. Suppose that after a deviation firms charge prices equal to their marginal costs for
τ periods and afterward return to collusion, and
• consumers search sequentially. Then the critical discount factor above which collusion is
sustainable decreases with the search cost.
• consumers search non-sequentially, and the search cost is such that in a competitive equilibrium a consumer samples all the firms. Then the critical discount factor above which collusion is
sustainable decreases with the search cost.
Since π d0 is lower than π ∗ , the critical discount factor δ0 is lower than δ ∗ . The difference π c − π d0
decreases with the search cost in both the sequential and the non-sequential search models. However,
this difference between profits is less sensitive to the search cost than the deviation gain, and both
δe0 and δe0 decrease if market transparency decreases.
6
Cartel stability with an exogenous fraction of uninformed
consumers
In this section, I show that by using a different method to model market transparency the result
in proposition 1 may be the opposite, i.e. cartel stability may decrease with market opacity. I
analyze a duopoly market in which a consumer has unit demand, and the valuations of products
are independently and identically distributed among consumers and varieties according to a uniform
distribution in the interval [0, 1]. There are no search costs in the model. The fraction φ of consumers
know their match values of both products but they do not observe the prices of the products without
visiting the firms, and they can visit only one seller. Meanwhile, 1 − φ consumers are fully informed.
It is said that market transparency decreases if φ goes up. An uninformed consumer expects that
both firms charge the same price, and she visits the firm where she gets a higher match value.
Irrespective of the possessed information, a consumer buys a product only if her match value of the
product is higher than the price.
I start with the derivation of the competitive equilibrium price p∗ . Suppose that firm i sets price
pi > p∗ , and firm j charges p∗ . Firm i sells to an uninformed consumer if εi > max {εj , pi }. The
´1
probability of this event equals pi εdε = 1/2 (1 − p2i ). The firm sells to an informed consumer if
12
εi − pi > max {εj − p∗ , 0}. The probability of this event equals
of firm i is
πi = p i
φ
1 − p2i + (1 − φ)
2
ˆ
´1
pi
(ε − pi + p∗ ) dε. Thus, the pay-off
1
∗
(ε − pi + p ) dε
pi
After taking the first-order conditions of πi and setting pi = p∗ , I get that in the competitive
equilibrium price equals
p
φ2 + 2 + φ − 1
p∗ =
2φ + 1
and every seller earns the same profit π ∗ :
p∗ 1 − (p∗ )2
.
π =
2
When the firms collude, they maximize the sum Π = π1 + π2 with respect to both prices. By
denoting the price of firm 1 by p1 and the price of firm 2 by p2 , I obtain that their joint pay-off is
∗
ˆ 1
φ
2
Π = p1
1 − p1 + (1 − φ)
(ε − p1 + p2 ) dε
2
p1
ˆ 1−p1
φ
2
+ p2
1 − p2 + (1 − φ) p1 − p2 +
(ε + p1 ) dε
2
0
After taking the first-order conditions and solving them for the joint profit-maximizing prices, it
√
appears that the colluding firms set the same price pc = 1/ 3, and the profit of one cartel member
√ −1
is π c = 3 3 .
Although collusion leads to a higher profit than competition, a firm can earn even more by
deviating from the joint price setting, provided that its competitor sets the collusive price. The
optimal deviation price that maximizes the deviation profit π d is
ˆ 1−pc
φ
d 2
c
d
c
p = arg max p
1− p
+ (1 − φ) p − p +
(ε + p ) dε
2
0
q
√ √
φ2 5 − 2 3 + φ 2 3 − 6 + 4 + 2φ − 2
=
3φ
d
d
The firms apply the standard grim trigger strategy. Hence the critical discount factor δ ∗ equals
−1
π d − π c π d − π ∗ . If an actual discount factor is above this threshold value, then a cartel is
sustainable. Otherwise, the firms never start colluding.
In Figure 2, δ ∗ is plotted as a function of the share of uninformed consumers. Contrarily to
the findings of Schultz (2005), the critical discount factor δ ∗ increases with the share of uninformed
consumers. The difference arises due to different distributions of match values in the two models. In
the set-up of Schultz (2005), the valuations of the products are negatively correlated, whereas they
13
are independently distributed in the random utility model. If the match values are uncorrelated,
then the share of consumers who draw relatively low valuations of a product and prefer it to the
other variety is bigger.10 Hence, the demand of a firm is more elastic to its price, and π ∗ is more
sensitive to the share of uninformed consumers than in Schultz (2005). This leads to the fact that,
differently from the set-up of Schultz (2005), in the random utility model, an increase in market
opacity has a stronger negative impact on the punishment than on the deviation gain and the critical
discount factor δ ∗ increases with φ.
The difference between the results of proposition 1 and Figure 2 is mainly driven by the different
sensitivity of the deviation gain to market transparency. In a search model, consumers sample
firms randomly. Therefore, there is a share of consumers whose first observed match value is lower
than the match value of a not-yet-searched variety. If the search cost increases, the share of these
consumers in the demand of the deviating firm is bigger, and it has a very strong negative effect
on the deviation profit. On the contrary, in the model with a share of uninformed consumers, an
uninformed consumer always goes to a firm where she gets the highest match value. Therefore,
the demand of the deviant is less price elastic, and the deviation profit is less sensitive to market
transparency than in the search models. As a result, the deviation gain in the model with the fraction
of uninformed consumers is less sensitive to the market transparency than the deviation gain in a
search model. Then the deviation gain decreases with market opacity less than the punishment and
the critical discount factor above which collusion is sustainable increases.
Figure 2: The values of δ ∗ for different φ. limφ→1 δ ∗ = 1/2.
7
Discussion
According to the International Competition Network (INC) “the prohibition against cartels is now
an almost universal component of competition laws” because competition between sellers is beneficial
for consumers and the “competitive process only works when competitors set prices independently.”11
10
It may happen that, in the random utility model, a consumer draws two low match values and one is higher than
the other. On the contrary, in the model of Schultz (2005), the valuations for the products are negatively correlated:
a low match value for one product implies a high match value for the other variety.
11
International Competition Network (2005).
14
On the contrary, firms prefer to engage in secret price (quantity) coordination because then they earn
higher profits. The objectives of firms and competition authorities contradict each other. Hence, it is
important for policy makers to determine the conditions which favor collusion in order to implement
proper consumer protection policy measures.
My findings show that lack of market transparency from the consumer point of view often facilitates collusion. Additionally, consumer surplus decreases with search costs in a competitive market.
Hence, competition authorities would prefer to increase market transparency from the consumer
point of view by promoting such measures as comparison websites and standardized product information. However, the studies on market transparency from the point of view of firms show that
cartels are less stable if the market is less transparent. By following the words of Stigler (1964),
“no one has yet invented a way to advertise price reductions which brings them to the attention of
numerous customers but not to that of any rival.” Therefore, an increase in market transparency
from the consumer point of view would necessarily increase market transparency from the point
of view of sellers, which would make collusion easier to sustain. As a result, before implementing
market transparency increasing actions in a particular market, it is important to determine which
of two effect on cartel sustainability is stronger.
15
References
Abreu, D. (1986). Extremal equilibria of oligopolistic supergames. Journal of Economic Theory 39,
191–225.
Abreu, D., D. Pearce, and S. Ennio (1985). Optimal cartel equilibria with imperfect monitoring.
Journal of Economic Theory 39, 251–269.
Anderson, S., A. de Palma, and J.-F. Thisse (1992). Discrete Choice Theory of Product Differentiation. The MIT press, Cambridge.
Anderson, S. P. and R. Renault (1999). Pricing, product diversity, and search cost: a BertrandChamberlin-Diamond model. RAND Journal of Economics 30 (4), 719–735.
Burdett, K. and K. L. Judd (1983). Equilibrium price dispersion. Econometrica 51 (4), 955–969.
Compte, O. (2002). On failing to cooperate when monitoring is private. Journal of Economic
Theory 102 (1), 151 – 188.
Green, E. J. and R. H. Porter (1984). Non-cooperative collusion under imperfect price information.
Econometrica 52 (1), 87–100.
Häckner, J. (1996). Optimal symmetric punishments in a Bertrand differentiated products duopoly.
International Journal of Industrial Organization 14, 611–630.
International Competition Network (2005). Defining hard core cartel conduct effective institutions
effective penalties.
Kandori, M. and H. Matsushima (1998). Private observation, communication and collusion. Econometrica 66 (3), pp. 627–652.
Kohn, M. G. and S. Shavell (1974, October). The theory of search. Journal of Economic Theory 9 (2),
93–123.
Moraga-González, J. L. and V. Petrikaitė (2013). Search costs, demand-side economies and the
incentives to merge under Bertrand competition. RAND Journal of Economics, 391–424.
Moraga-González, J. L., Z. Sándor, and M. R. Wildenbeest (2013, March). Do higher search costs
make the markets less competitive?
Nilson, A. (1999, December). Transparency and competition. Stockholm School of Economics
Working Paper No. 298.
Perloff, J. M. and S. C. Salop (1985). Equilibrium with product differentiation. Review of Economic
Studies 52 (1), 107–120.
16
Schultz, C. (2005). Transparency on the consumer side and Tacit collusion. European Economic
Review 49 (2), 279–297.
Stigler, G. J. (1964). A theory of oligopoly. The Journal of Political Economy 72 (1), 44–61.
Wolinsky, A. (1986). True monopolistic competition as a result of imperfect information. Quarterly
Journal of Economics 101 (3), 493–512.
17
Appendix A
The changes of (3) with k.
The second derivative of (3) with respect to k is
h i
1
2
2
∗ k+1
∗ k+1
∗
∗ k+1
∗
2
(p
)
−
1
−
2
(k
+
1)
(p
)
ln
p
+
(k
+
1)
(p
)
ln
(p
)
(k + 1)3
The expression increases in p∗ as its derivative with respect to p∗ is (p∗ )k ln2 (p∗ ) > 0. Thus the
expression is less than if by taking p∗ → 1. After taking the limit the expression equals −2/ (k + 1)3 ,
which implies that (3) is concave in k.
The first derivative of (3) with respect to k is
1 − (p∗ )k+1 + (p∗ )k+1 ln (p∗ )k+1
−s
(k + 1)2
The first term decreases in k and limk→∞
1−(p∗ )k+1 +(p∗ )k+1 ln(p∗ )k+1
(k+1)2
= 0.
Proof of proposition 1.
After solving the first order conditions for an optimal deviation price I get that ped and pbd can be
written as follows
1 1 − x̄
d
c n
c
pe =
(1 − (e
p ) ) + pe
2 1 − x̄n
1 c 1
c k
d
1 − (b
p)
pb +
pb =
2
k
h
i
h
i
Sequential search. Firstly, I observe that sign ∂ δe∗ /∂s = −sign ∂ δe∗ /∂ x̄ . Secondly, the profit of
a coalition member does not depend on the search cost. Thus, the derivative of the critical discount
factor with respect to x̄ depends only on the changes of π
ed and π
e∗ .
d
d
de
π
dδe∗
1
de
π
de
π∗
d
∗
d
c
=
π
e −π
e −
−
π
e −π
e
dx̄
dx̄
dx̄
(e
πd − π
e∗ )2 dx̄
d
1
de
π
de
π∗ d
c
∗
c
=
(e
π −π
e )+
π
e −π
e
dx̄
(e
πd − π
e∗ )2 dx̄
(19)
The denominator of the first fraction in (19) is positive. Hence,
"
dδe∗
sgn
dx̄
#
de
πd c
de
π∗ d
∗
c
= sgn
(e
π −π
e )+
π
e −π
e
dx̄
dx̄
As ∂ pec /∂ x̄ = 0 and ∂e
π d /∂ ped = 0 then
18
(20)
de
πd
ped 1 + (n − 1) x̄n − nx̄n−1 c
d
p
e
−
p
e
=
dx̄
n
(1 − x̄)2
The competitive equilibrium profit may be written as follows
∂qj (e
p∗ )2 1 − x̄n
π
e = (e
p)
=
∂pj pj =ep∗
n 1 − x̄
∗ 2
∗
Then, by using the Implicit function theorem for ∂ pe∗ /∂ x̄, the derivative of the competitive profit
with respect to x̄ may be written as
∂e
π∗
(e
p∗ )2 1 + (n − 1) x̄n − nx̄n−1
=
∂ x̄
n
(1 − x̄)2
n
1−
!
2 1−x̄
1−x̄
1−x̄n
1−x̄
+ n (e
p∗ )n−1
Note that both the derivative of π
ed with respect to x̄ and the derivative π
e∗ with respect to x̄
n
n−1
−nx̄
have the same positive element n1 1+(n−1)x̄
. The sign of the RHS of (20) does not change if I
(1−x̄)2
divide it by this expression. Therefore, I explore the sign of the expression φ (x̄, pe∗ , n) further on.
n
c
c n
∗ 2 1 − x̄
φ (x̄, pe , n) ≡ pe pe − pe
pe (1 − (e
p ) ) − (e
p)
1 − x̄
!
1−x̄n
n
2 1−x̄
∗ 2
c
c n
d 2 1 − x̄
+ (e
p ) 1 − 1−x̄n
− pe (1 − (e
p) )
pe
1 − x̄
+ n (e
p∗ )n−1
1−x̄
∗
d
c
d
By using the expression of ped and the fact that 1 − (pc )n = n (pc )n I get
d
n π
e −π
e
c
=
1 1 − x̄n
4 1 − x̄
=
1 1 − x̄n
4 1 − x̄
!
2
1 − x̄
1 − x̄
1 − x̄n
c n+1
c 2
2
c 2n
n
(e
p
)
+
− n (e
pc )n+1
(e
p ) +2
n
(e
p
)
n
n
1 − x̄
1 − x̄
1 − x̄
!
2
1 − x̄
1 − x̄
(e
pc ) 2 − 2
n (e
pc )n+1 +
n2 (e
pc )2n
n
n
1 − x̄
1 − x̄
2
2
1 − x̄
1 − x̄n c
c n
pec −
n
(e
p
)
=
pe − ped
n
1 − x̄
1 − x̄
1
=
4
The collusive price is higher than the deviation price. Therefore, if I divide φ (x̄, pe∗ , n) by pec − ped
then the new expression will have the same sign as φ (x̄, pe∗ , n). I denote this new expression by
φ1 (x̄, pe∗ , n)
n
∗
d
c n+1
∗ 2 1 − x̄
φ1 (x̄, pe , n) = pe n (e
p)
− (e
p)
+ (e
p∗ )2
1 − x̄
From the first order condition (6) it is true that
19
n
1−
2 1−x̄
1−x̄
1−x̄n
1−x̄
+ n (e
p∗ )n−1
!
pec − ped
1 − x̄n
1 − x̄
∗ n−1
(e
p)
1
1
1 − x̄n
= ∗−
= ∗ −γ
pe
1 − x̄
pe
Then I can rewrite φ1 (x̄, pe∗ , n) as
∗
c n+1
d
p)
φ1 (x̄, pe , n) ≡ pe n (e
∗ 2
∗ 2
− (e
p ) γ + (e
p)
n − pe∗ γ (n + 1)
n − γ (n − 1) pe∗
pec − ped γ
Now note that
1 ∂ (φ1 (x̄, pe∗ , n))
= −e
pd −
2
∗
∂γ
(e
p)
n − pe∗ γ (n + 1)
2ne
p∗ γ
−
(n − γ (n − 1) pe∗ )2 n − γ (n − 1) pe∗
The inequality has been obtained because
Additionally,
∂e
π∗
∂ x̄
< 0 implies that
∂ (φ1 (x̄, pe∗ , n))
= −2e
pd pe∗ γ + 2e
p∗
∂ pe∗
− 2 (e
p∗ )2
n−e
p∗ γ(n+1)
n−γ(n−1)e
p∗
pec − ped < 0
< 0.
n − pe∗ γ (n + 1)
pec − ped γ
∗
n − γ (n − 1) pe
γ 2n
<0
pec − ped
(n − γ (n − 1) pe∗ )2
and
∂ (φ1 (x̄, pe∗ , n))
= n (e
πc − π
e∗ ) − (e
p∗ ) 2 γ
∂ ped
n − pe∗ γ (n + 1)
n − γ (n − 1) pe∗
> 0,
and the fact ∂ ped /∂ x̄ < 0 implies that ∂ ped /∂γ < 0 Therefore,
c
1 c
1
pe n
n
1 n−
1
pe +
−
+
1, , n =
2
2
n+1
n+1 4
8 n−
n
2
n
2
(n + 1)
φ1 (x̄, pe , n) > φ1
(n − 1)
1
2
n 2 (n − 3) + (n − 1) (5 + n) (n + 1) n −1 − (1 + n) n (n − 2)
=
4 (n − 3) (n + 1)1+2/n
∗
pec −
1
n+1
n
(21)
limn→3 φ1 1, 21 , n = ∞ and the expression is positive for all n > 3 (see Figure 3a)
∗
Now I tackle the case n = 2 by using the fact that γ = 1 − (e
p∗ )2 /e
p . If n = 2 then
√ ∗
1 − 3 (e
p∗ ) 2 φ1 (x̄, pe , 2) =
−3
+
7
3e
p − 9 (e
p∗ )2
4
∗
27 (e
p ) −1
√
√
∗ 3
∗ 4
∗ 5
−8 3 (e
p ) + 12 (e
p ) + 3 3 (e
p)
∗
The fraction
1−3(e
p∗ )2
27((e
p∗ )4 −1)
is negative. The polynomial in parenthesis is also negative (See Figure
20
3b).
As φ1 > 0 then δe∗ increases in x̄ or decreases in s.
(a)
(b)
Figure 3: The values of φ1 for different values of pe∗ and n
Non-sequential search.
h
i
∗
b
If the search costs are such that a consumer samples all the firms in the market then sign ∂ δ /∂s =
h
i
−sign ∂ δb∗ /∂k . For any k
k+1
k
π
bc = (k + 1)− k
n
k pbd
π
b =
n
2
d
k
=
4n
1
(k + 1)− k +
1
k+1
2
nb
π ∗ = pb∗ (1 − (b
p∗ )n )
where pb∗ is defined by
δb∗ =
k
4
1 − np∗ − (p∗ )n = 0
2
2
1
1
k
− k1
1
(k + 1)− k + k+1
−
k+1
(k
+
1)
−
k+1
(k+1) k
=
2
2
1
1
k
1
1
(k + 1)− k + k+1
(k + 1)− k + k+1
− nb
π∗
− k4 nb
π∗
4
k+1
2
g (nb
π ∗ , k) ∂ δb∗
2
∗
k
k
= k (k − 1) nb
π (k + 1)
− k (k + 1) − (k + 1)
1
4(1 + k) k ∂k
1
2
+ k (k + 1)2 − 2b
π ∗ n (k + 1)3+ k + (k + 1) k 2b
π ∗ n (k + 1)2 − k
× ln (k + 1)k+1
(22)
21
2
1
Figure 4: The value of (k 2 + 1) (k + 1) k − (k + 1) k +3 + 2k(k + 1)2 for different k
where
1
1
∗
g (nb
π , k) = k (1 + k)2 + 2k (k + 1)1+ k
k
2
2
2
∗
k
+ (k + 1) k − 4b
π (k + 1)
>0
The expression on the RHS of (22) is linear in nπ ∗ and decreases in nπ ∗ because its derivative
with respect to nπ ∗ is negative:
2
1
(k + 1)1+ k k (k − 1) (k + 1) k − (k + 1)2 +
1
1
(k + 1)1+ k 2 (k + 1)2 ln (k + 1) (k + 1) k − 1 − k < 0
Therefore, if the RHS of (22) is positive for the highest value of nb
π ∗ then it is negative for all
reasonable nb
π ∗ . max {nb
π ∗ } = 1/4. Thus, the RHS of (22) is more than
2
1
1
k+1
+3
2
2
k
k
+ 2k(k + 1) +
ln (k + 1)
k + 1 (k + 1) − (k + 1)
2
k+1
2
1
k (k − 1)
(k + 1) k − k
(k + 1) k − (k + 1)2
4
k+1
2
(23)
2
Note that 41 (k + 1) k − k < 12 (k + 1) − k < 0, (k + 1) k − (k + 1)2 < 0 and (k 2 + 1) (k + 1) k −
1
(k + 1) k +3 + 2k(k + 1)2 > 0 (See Figure 4). Hence, ∂ δb∗ /∂k > 0 or ∂ δb∗ /∂s < 0.
Proof of proposition 2
Sequential search
"
∂ δe0
sgn
∂ x̄
#
∂e
∂e
πd c
π d0 d
d0
c
= sgn
π
e − πs +
π
e −π
e
∂ x̄
∂ x̄
22
By setting pe∗ = 0 in (5) and solving for an optimal deviation price I get
ped0 =
and
1 − x̄
,
2 (1 − x̄n )
π
ed0 =
1 − x̄
4n (1 − x̄n )
∂e
π d0
−1 + nx̄n−1 − (n − 1) x̄n
=
<0
∂ x̄
4n (1 − x̄n )2
Then after plugging in the values of the derivatives and profit I get that
d
∂πsd0 d
∂e
π
n2 (1 − x̄)2
c
d0
c
=
π
e −π
e +
π
e −π
e
(e
pc − ped ) (1 + (n + 1) x̄n − nx̄n−1 ) ∂ x̄
∂ x̄
pec
1 − x̄
1
c n+1
2
c 2n
n (e
p)
+
n (e
p) −
2
1 − x̄n
2
pc )2n − 1/2 = n2 / (n + 1)2 − 1/2 > 0. Therefore,
If n ≥ 3 then n2 (e
c n+1
n (e
p)
1 − x̄
+
1 − x̄n
1
1 − pec
1
2
c 2n
c n+1
2
c 2n
n (e
p) −
≥ n (e
p)
+
n (e
p) −
2
1 − (e
pc )n
2
c
n
(n + 1) (1 − pe )
=
−
n+1
2ne
pc
(n + 1) 1 − 12
n
−
>
n+1
2n 12
n
n+1
n2 − 2n − 1
=
−
=
≥0
n+1
2n
2n (n + 1)
If n = 2 then
c n+1
n (e
p)
1 − x̄
+
1 − x̄n
2
1
2
1
1
2
c 2n
≥ √ −
>0
n (e
p) −
= √ −
2
3 3 18 (1 + x̄)
3 3 36
As a result, ∂ δe0 /∂ x̄ > 0 or ∂ δe0 /∂s < 0.
Non-sequential search
If a consumer samples all n firms in a competitive equilibrium then she does it when all prices
are zero. After I solve the profit maximization problem of the firm I get
pbd0 =
1
,
n+1
π
bd0 =
nn−1
(n + 1)n+1
The expression of δb0 is similar to δb∗ except for π ∗ . Thus, to prove that δb0 increases with k I need
to prove that the RHS of (24) is positive.
23
k+1
2
g (nb
π ∗ , k) ∂ δb0
2
d0
k
k − (k + 1)
=
k
(k
−
1)
nb
π
(k
+
1)
−
k
(k
+
1)
1
4(1 + k) k ∂k
2
2
k+1
2
3+ k1
d0
d0
k
π n (k + 1) − k
+ (k + 1) 2b
+ ln (k + 1)
k (k + 1) − 2b
π n (k + 1)
(24)
If k = n then pb∗ > pbd0 . This can be seen by plugging pbd0 in the LHS of (9), which gives that the
LHS of (9) is positive: (n + 1)−1 − (n + 1)−n > 0 . Then,
1
π
b >
(n + 1) n
∗
1−
1
(n + 1)n
=
(n+1)n −1
n
n+1
(n + 1)
>π
bd0
. By using the arguments from the proof of proposition 1 I get that ∂ δb0 /∂k > 0 or ∂ δb0 /∂s < 0.
24
Appendix B. Concavity of the cartel pay-off function
Letting the cartel to choose an arbitrary set of n prices makes the derivation of the joint pay-off
functions very complicated. The difficulty arises when all possible search orders and permutations of
firms have to considered while deriving the general joint pay-off function. Therefore, here I provide
the proof of local concavity for the case when n = 2 or n = 3.
Sequential search
Suppose there are two firms in the market and they choose the set of collusive prices {p1 , p2 } =
6
{e
pc , pec }. With probability 1/2a consumer goes to firm 1 first and later may visit the second firm.
With probability 1/2 the consumer may visit the second firm first and later may search in the first
firm. Thus, the pay-off of a cartel is
ˆ x̄−epc
1
c
Π = p1 1 − (x̄ − pe + p1 ) +
(ε + p2 ) dε
2
0
ˆ x̄−epc
1
c
c
+ p2 (x̄ − pe + p1 ) (1 − (x̄ − pe + p2 )) +
(ε + p1 ) dε
2
0
ˆ x̄−epc
1
c
c
+ p1 (x̄ − pe + p2 ) (1 − (x̄ − pe + p1 )) +
(ε + p2 ) dε
2
0
ˆ x̄−epc
1
c
+ p2 1 − (x̄ − pe + p2 ) +
(ε + p1 ) dε
2
0
The pair of a joint profit maximizing prices is determined by the following system of first order
conditions
1
1
(1 + x̄ − pec + pj ) (1 − x̄ + pec − 2pi ) + pj (1 − pj )+
2
2
ˆ x̄−epc
1
(ε + pj ) dε = 0 ∀ {i, j} = {1, 2} , i 6= j
+ pj (x̄ − pec ) +
2
0
The elements of a Hessian matrix are
∂ 2Π
Hii =
= − (1 + x̄ − pec + pj ) < 0
∂pi ∂pi
Hij =
∂ 2Π
1
= (2 + x̄ − pec − 2pi − 2pj )
∂pi ∂pj
2
If Hij is positive then |Hii | > |Hij |because
1 + x̄ − pec + pj −
1
(2 + x̄ − pec − 2pi − 2pj ) =
2
1
1
x̄ − pec + pi + 2pj > 0
2
2
25
Additionally, if Hij is negative then still |Hii | > |Hij |because
1 + x̄ − pec + pj +
1
(2 + x̄ − pec − 2pi − 2pj ) =
2
2 + 2x̄ − 2e
pc − pi > 0
Hence, |∂pi/∂pj | < 1, which implies that the pair of prices (p1 , p2 ) which satisfies the system of
the first order conditions is unique. One can easily check that the first order conditions are satisfied
√
√
if p1 = p2 = pec = 1/ 3. Finally, at the point p1 = p2 = pec = 1/ 3 the determinant of the Hessian is
positive:
5
5x̄
3
25
+ √ >0
|H||p1 =p2 =epc =1/√3 = x̄ + x̄2 + √ −
4
3 12 2 3
Therefore, the symmetric solution gives a local maximum of the function.
Non-sequential search
For the analysis of the set of the joint profit maximizing prices I address the case when n = 3
and consumers sample two firms. Suppose that a cartel contemplates setting the set of prices
{p1 , p2 , p3 } =
6 {b
pc , pbc , pbc } and p1 < p2 < p3 Any pair of firms is sampled with probability 1/3. Thus,
the payoff of the cartel is
ˆ 1−p3
ˆ 1
p1 p3 − p1 +
(ε + p3 ) dε + p3
(ε − p3 + p1 ) dε
0
p3
ˆ 1−p2
ˆ 1
1
p1 p2 − p1 +
(ε + p2 ) dε + p2
+
(ε − p2 + p1 ) dε
3
0
p2
ˆ 1−p3
ˆ 1
1
p2 p3 − p2 +
(ε + p3 ) dε + p3
(ε − p3 + p2 ) dε
+
3
0
p3
1
Π=
3
The system of the first order conditions is as follows
3 2
3 2
2p3 − 4p1 + 1 − p3 + 2p2 − p2 = 0
2
2
1
3 2
3 2
1 + 2p1 − 3p2 p1 − 4p2 + p2 + 2p3 − p3 = 0
3
2
2
1
1 + 2p1 − 3p1 p3 − 4p3 + 3p23 + 2p2 − 3p2 p3 = 0
3
1
3
(25)
(26)
(27)
From (25) I express the value of p1 and plug it in (26) and (27). As a result the pair (p2 , p3 ) that
satisfies the system (25)-(27) must satisfy the following equations
4p2 − 2 + 3p22 − 4p3 + 3p23 = 0
(28)
4 + p2 (8 − 12p3 ) − 10p2 + 2p23 + 3p33 − p22 (2 − 3p3 ) = 0
(29)
26
From (28) I get that
1
p2 =
3
q
2
−2 + 10 + 12p3 − 9p3
After plugging this value in (29) and rearranging the terms I get the equation
9p23 − 5 − (6p3 − 4)
q
10 + 12p3 − 9p23 = 0
√
After solving the equation the only root which belongs to the interval [0, 1] is p3 = pbc = 1/ 3. After
computing the values of p2 and p1 i get that the solution of (25)-(27) is p1 = p2 = p3 = 1/3.
The Hessian matrix elements are
4
H11 = − ,
3
H21 =
1
(2 − 3p2 ) ,
3
H12 =
1
(2 − 3p2 ) ,
3
1
H22 = − (3p1 + 4 − 3p2 ) ,
3
1
1
(2 − 3p3 ) , H32 = (2 − 3p3 )
3
3
At the solution point the matrix becomes
H31 =


H=
As
4
|H11 | = − < 0,
3
H13 =
−4
√3
2− 3
3
− 34
√
2− 3
3√
2− 3
3
√
2− 3
3
− 34
√
2− 3
3
− 34
√
2− 3
3
1
(2 − 3p3 )
3
H23 =
1
(2 − 3p3 )
3
1
H33 = − (3p1 + 4 − 6p3 + 3p2 )
3
√
2− 3
3√
2− 3
3
− 43



4
= 1 + √ > 0,
3 3
√
8 3 − 26
√
|H| =
<0
3 3
I conclude that the symmetric solution give the local maximum of the joint pay-off function.
27