In this note we characterize optimal punishments with detection lags when the market consists of n oligopolistic firms. We extend a previous note by Colombo and Labrecciosa (2006) [Colombo, L., and Labrecciosa, P., 2006. Optimal punishments with detection lags. Economic Letters 92, 198-201] to show how in the presence of detection lags optimal punish- ments fail to restore cooperation also in markets with a low number of firms.optimal punishments, detection lags, collusion sustainability

Ciarreta Antuñano, Aitor

Gutiérrez Hita, Carlos

English

Research Papers in Economics

ISSN 1988-088X 
 
 
 
  
 Department of Foundations of Economic Analysis II 
 University of the Basque Country 
 Avda. Lehendakari Aguirre 83 
 48015 Bilbao (SPAIN) 
 http://www.ehu.es/FAEII 
 
 
 
DFAE-II WP Series 
 
 
Aitor Ciarreta & Carlos Gutiérrez-Hita 
 
A note on collusion sustainability with optimal 
punishments and detection lags 
2008-08 A note on collusion sustainability with optimal
punishments and detection lags￿
Aitor Ciarretaa and Carlos GutiØrrez-Hitab,y
aUniversidad del Pa￿s Vasco, Bilbao, Spain.
bUniversitas Miguel HernÆndez, Elche, Spain.
September 5, 2008
Abstract
In this note we characterize optimal punishments with detection lags
when the market consists of n oligopolistic ￿rms. We extend a previous
note by Colombo and Labrecciosa (2006) [Colombo, L., and Labrecciosa,
P., 2006. Optimal punishments with detection lags. Economic Letters 92,
198-201] to show how in the presence of detection lags optimal punish-
ments fail to restore cooperation also in markets with a low number of
￿rms.
Keywords: Optimal punishments; Detection lags; Collusion sustain-
ability.
JEL Classi￿cation: C73; D43
1 Introduction
Strategic interaction plays an important role to determine the sustainability of
cartel agreements. Friedman (1971) stressed the importance of time for col-
lusion sustainability under trigger strategies pro￿le. Abreu (1986, 1988) also
characterized optimal punishments in supergames. He shown that a symmetric
optimal penal code yields the lowest critical discount factor such that collusion
can be sustained. Moreover, Abreu et al. (1986) and Fudenberg and Maskin
(1986) characterize folk theorems for in￿nitely repeated games with discounting.
￿Financial support from Ministerio de Ciencia y Tecnolog￿a (SEJ2006-06309), and Gob-
ierno Vasco (IT-313-07) is gratefully acknowledged. The usual disclaimers applied.
yCorresponding author. Mailing address: av/Univesidad s/n, 03202, Elche, Spain. Tel.:
+34 966658869; fax: +34 966 658564. E-mail address: cgutierrez@umh.es.
1Colombo and Labrecciosa (2006) describe in a previous note an optimal penal
code with n detection lags in duopoly supergames. They attemp to show how
large detection lags after deviation hinders collusion. They only provide exam-
ples with one period detection lag under Bertrand and Cournot competition.
This note characterizes collusion sustainability with detection lags in a n-￿rm
oligopoly. We extend the result by Colombo and Labrecciosa (2006) showing
how optimal punishments fail to restore cooperation for a su¢ ciently large de-
tection lag. We show that there is a trade-o⁄ between the amount of detection
lags and the maximum number of ￿rms compatible with collusion sustainability.
2 The model
Consider an in￿nitely repeated n-￿rm game (n ￿ 2) with discounting ￿. Call
such game Gl (N;1;￿) where the superscript denotes that deviation is detected
after l period(s); that is, we assume that an individual deviation from any
collusive agreement is detected by the rest n￿1 ￿rms after l period(s). Following
Colombo and Labrecciosa (2006) ￿rms agree upon the following penal code: If
a deviation occurred at any given time ￿, it is detected at the end of the period
￿ +l, then ￿rms adopt the punishment strategy symmetrically at time ￿ +l+1;
￿nally, they restore cooperation from period ￿ + l + 2 ownwards. If, otherwise,
any ￿rm at time ￿ + l + 1 does not join the penal code and decides not to
retaliate, the punishment phase continues until the same action is taken by the
n ￿rms.
We call the best collusive strategy s￿ = (s￿
1;::;s￿
i;::;s￿
n), and the optimal
punishment strategy sc = (sc
1;::;sc
i;::;sc
n). ￿d (s￿) are deviation pro￿ts from
the best collusive pro￿ts ￿ (s￿), ￿ (sc) are the pro￿ts during the punishment
phase, and ￿d (sc) are optimal deviation pro￿ts from the punishment phase.
Then, following Abreu￿ s theorem 15 and Colombo and Labrecciosa￿ s (2006)
expressions (1) and (2), we state that,
Proposition 1 Let (s￿;sc) be the optimal stick-and-carrot punishment with an
l period detection lag. Then the following is hold,
l X
t=0
￿
t ￿
￿d (s￿) ￿ ￿ (s￿)
￿
￿ ￿
l+1 [￿ (s￿) ￿ ￿ (sc)]; (1)
l X
t=0
￿
t ￿
￿d (sc) ￿ ￿ (sc)
￿
=
l+1 X
t=1
￿
t [￿ (s￿) ￿ ￿ (sc)]; (2)
where l is the detection lag.1
1When l = 0 and the assumption that s￿ = sm where sm is the monopoly outcome,
theorem 15 of Abreu (1986, p.203) is obtained. Notice also that Proposition 1 by Colombo
and Labrecciosa (2006) should be applied only in the case of zero detection lags.
2Proof. Consider ￿rst expression (1) by Colombo and Labrecciosa (2006).
Rewritting
Pl
t=0 ￿ (s￿) into
Pl
t=0 ￿ (s￿) +
P1
t=l+1 ￿ (s￿) we obtain,
l X
t=0
￿
t ￿
￿d (s￿) ￿ ￿ (s￿)
￿
￿
1 X
t=l+1
￿
t￿ (s￿) ￿
1 X
t=l+2
￿ (s￿) ￿ ￿
l+1￿ (sc);
where the left hand side reach the ones at our equation (1). Now, taking
the terms
P1
t=l+1 ￿
t￿ (s￿) ￿
P1
t=l+2 ￿ (s￿) on the right hand side we obtain
￿
l+1￿ (s￿) which jointly with ￿
l+1￿ (sc) ￿ts the right hand side of equation (1)
in Proposition 1. Now, consider expression (2) by Colombo and Labrecciosa
(2006). After adding ￿
Pl+1
t=1 ￿
t￿ (s￿) in both sides and reordering we obtain
Pl
t=0 ￿
t￿d (sc)￿
Pl+1
t=1 ￿
t￿ (sc)￿￿ (sc)+￿
l+1￿ (sc)=
P1
t=1 ￿
t￿ (s￿)￿
P1
t=l+2 ￿
t￿ (s￿)￿
Pl+1
t=1 ￿
t￿ (sc):
With the last three terms on the left hand side we can obtain
Pl
t=0 ￿
t￿ (sc).
Then, adding it to the ￿rst term we get the left hand side on equation (2) in
Propostion 1. Finally, take
P1
t=1 ￿
t￿ (s￿) ￿
P1
t=l+2 ￿
t￿ (s￿) on the right hand
side to obtain
Pl+1
t=1 ￿
t￿ (s￿). It must be added to
Pl+1
t=1 ￿
t￿ (sc) to get the
desired expression on the right hand side of equation (2). This completes the
proof.
Equation 1 is the no-defection condition. Equation 2 requires that each ￿rm
does not deviate from the punishment path after deviation is detected. If that
was the case, punishment would follow until all ￿rms agree to go along.
We now consider the game Gl (N;1;￿) to ￿nd how detection lags and
the number of ￿rms a⁄ect collusion sustainability. Market demand is given
by p(Q) = 1 ￿ Q where Q =
PN
i=1 qi. We assume that marginal cost are
zero for each ￿rm.2 Firms engage in quantity competition. Then, a strategy
si = qi. By using Proposition 1 and the best response function qi(qc
￿i) = ￿
1 ￿ c ￿ (n ￿ 1)qc
￿i
￿
=2 where qc
￿i is ths punishment strategy of the remaining
n ￿ 1 ￿rms, equations (1) and (2) can be expressed as,
Pl
t=0 ￿
t
h
(n+1)
16n2 ￿ 1
4n
i
= ￿
l+1 ￿ 1
4n ￿ (1 ￿ nqc
i)qc
i
￿
;
Pl
t=0 ￿
t ￿
(1 ￿ qi(qc
￿i) ￿ nqc
￿i)qi(qc
￿i) ￿ (1 ￿ nqc
i)qc
i
￿
=
Pl+1
t=1 ￿
t ￿ 1
4n ￿ (1 ￿ nqc
i)qc
i
￿
:
We proceed as follows: for a given number of detection lags ￿nd the maxi-
mum number of ￿rms compatible with a discount factor ￿ < 1. Corollary 1
summarizes the results.
Corollary 1 For the game Gl (N;1;￿) when optimal punishments ￿ la Abreu
are adopted, under Cournot competition collusion is sustainable accordingly the
2This assumption allows for negative prices during the punishment phase. This is done for
ease of exposition.
3following trade-o⁄ between number of lags and number of ￿rms,
Table 1. Detection lags and number of ￿rms
# lags 1 2 3;4 5;6;7 8;::;14 15;16 17;::;270
# firms 9(:96) 7(:94) 6(:98) 5(:99) 4(:99) 3(:94) 2(:999)
Each number in parenthesis in Table 1 is the minimum value of the discount
factor such that collusion is sustainable for the largest detection lag compatible
with the number of ￿rms considered.3 For example, with seven detection lags
￿ve ￿rms are able to sustain collusion over time i⁄ ￿ 2 (:99;1). It is shown that
there is an inverse relationship between the amount of detection lags and the
number of ￿rms that reach collusion sustainability over time.
3 Concluding remarks
We consider an in￿nitely repeated game with discounting to extend the result by
Colombo and Labrecciosa (2006). We show also that when quantity competition
is considered, given a number of ￿rms, Abreu￿ s stick-and-carrot punishment fails
to restore cooperation for a su¢ cient large detection lag. This result calls for
attention to detect collusion in markets characterized by infrequent interaction
or imperfect information. We prove that the sustainability of collusion could be
di¢ cult also if the number of ￿rms is low.
References
[1] Abreu, D.J., 1986, Extremal equilibria of oligopolistic supergames, Journal
of Economic Theory 39, 191-225.
[2] Abreu, D.J., 1988, On the theory of in￿nitely repeated games with discount-
ing, Econometrica 56, 383-396.
[3] Abreu, D.J., D. Pearce and E. Stacchetti, 1986, Optimal cartel equilibria
with imperfect monitoring, Journal of Economic Theory 39, 251-269.
[4] Colombo, L. and P. Labrecciosa, 2006, Optimal punishments with detection
lags, Economics Letters 92, 198-201.
[5] Friedman, J.W., 1971, A non-cooperative equilibrium for supergames, Re-
view of Economic Studies 28, 1-12.
[6] Fundenberg, D. and E. Maskin, 1986, The folk theorem in repeated games
with discounting or with incomplete information, Econometrica 54, 533-554.
3Upon request authors send the programme which provides the results reported. Calcula-
tions have been run with the computer package Mathematica R ￿ by Wolfran Research.
4

A note on collusion sustainability with optimal punishments and detection lags

https://addi.ehu.es/bitstream/10810/6611/1/wp2008-08.pdf

A note on collusion sustainability with optimal punishments and detection lags

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics