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Sustaining collusion when the choice of strategic variable is endogenous
JOURNALOF
Journal of Economic Behavior and Organization Vol. 28 (1995) 373-385
EconomicBehavior & Oganization
Sustaining collusion when the choice of strategic variable is endogenous R. Rothschild Lancaster University, Department of Economics, Lancaster LA1 4Ex, UK
Received 14 February 1994; revised 29 August 1994
Abstract This paper addresses the problem of maintaining collusion in a differentiated duopoly when both the prospective deviant and punisher are free to choose between price and output as their strategic variables. We show that the traditional formulation of the supergame in which fiis commit for the duration to either price or output as their single common strategic variable overlooks the possibility of an asymmetric equilibrium in the ‘punishment’ phase. The results obtained here have obvious parallels with those found in the comparative statics literature. JEL classification: Keywords:
D43
Collusion; Supergame; Punishment; Price and output strategies
1. Introduction
In recent years the literature of oligopoly theory has been dominated by the search for mechanisms which will maintain the integrity of collusive arrangements in the face of the potential for deviant behaviour. The best known examples are due to Friedman (1971) and Abreu (1986). The underlying principle is that firms agree to adhere to a particular price or output level for as long as their fellow oligopolists do the same, but either revert to noncooperative behaviour forever in the event of deviation from this agreement (Friedman), or adopt a ‘carrot and stick’ rule which encourages even more severe punishments of finite duration followed by a return to cooperation (Abreu). In these models if, for any prospec0167-2681/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2681(95)00041-O
374
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of Economic Behavior & Org. 28 (1995) 373-385
tive deviant, the one-period gain from deviation is less than the (discounted) value of the losses arising from subsequent punishment, the cartel price or output will be maintained. Although the credibility of the threat which under-pins such strategies has been the subject of some discussion [see, for example, Rothschild (1985) and Al-Nowaihi and Levine (199211 the concept of the ‘trigger strategy’ has now become widely accepted. In light of this fact it seems appropriate to consider a neglected issue. This concerns the implicit assumption in the earlier literature that if price (output) is selected initially by the coalition as the strategic variable, then any deviation were it to occur - and any subsequent ‘punishment’ will be implemented in terms of that variable. In other words, the threat strategy requires that there be no switching by any firm between the two strategic variables once the initial choice has been made. The requirement that the variable be exogenously given in this way seems both unreasonable and restrictive, especially when one considers markets in which production capacity is highly flexible. For example, in their analysis of a related problem, Klemperer and Meyer (1986) consider the case of a consultant who chooses between a fee per hour and a fixed fee for a project as a whole. By charging per hour the consultant is setting a price; in charging for the project as a whole he or she has the option of spending less time on the client when the office is busy than when it is not (so that the work day is constant) and is in this sense setting a quantity and adjusting price to demand. More familiar examples of this phenomenon can be found in commercial aviation and in the oil industry, in which firms may compete either in price or in output. For instance, it is well known that airline operators have the option of either cutting price per seat on a given route or increasing the number of seats on offer at the prevailing price. Similarly, oil producers may elect to either undercut their rivals in price or ‘cheat’ on an agreement by increasing their output beyond quota. In markets of this type sellers can, moreover, switch easily between the price and output variables, and it is difficult to conceive of any mechanism which will prevent such switching if it is in the interests of agents to do so. In this paper we consider possible outcomes when the deviant is able to select a strategic variable different from that chosen initially by the coalition, and the ‘loyal’ member, given this selection, is similarly in a position to choose for the punishment phase the variable which yields to it the highest profits available. In focusing on the problem in the context of a supergame this paper goes beyond a related literature which has thus far been restricted to the comparison of Bertrand and Cournot behaviour [see, for example. Hathaway and Rickard (1979), Singh and Vives (19841, and Cheng (1985)]. The emphasis in these latter contributions has been largely on comparing the associated equilibria of the models (in terms of price, output and welfare) in a static framework. Once the constraints imposed by this framework are relaxed, it becomes possible to consider explicitly the implications of ‘switching’. In particular, we wish in this paper to consider the possibility that (a) the firm
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ofEconomic Behavior & Org. 28 (1995) 373-385
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which undertakes the punishment will be able to choose between the price and output variables for the ‘punishment’ phase, in order if necessary to threaten the most effective response (in terms of the power to deter) to deviant behaviour, and (b> the deviant will select for itself both the most profitable form of deviation and the most appropriate response to the punishment variable which the punisher threatens to choose. In addressing these questions we shall show, inter alia, that when firms are able to choose freely between the price and output variables, the preferred choice of the deviant will be price for the initial deviation, followed by output for the punishment. By contrast, however, we shall show also that while subgame perfection may dictate that in this case the punisher also chooses output for the punishment phase, the prospect of a ‘more severe’ punishment involving an asymmetric equilibrium might constitute a more effective deterrent to deviation. The paper is organised as follows. Section 2 sets out the model. Section 3 considers the implications of permitting firms to choose freely between the price and output variables. Section 4 offers some concluding comments. An Appendix sets out the two ‘benchmark’ cases of Bertrand and Cournot competition which will serve as a basis for the discussion in Section 3. Some simplifying notation is also introduced.
2. The model We consider a duopoly in which both firms i,i = 1,2, produce their output qi subject to identical, constant marginal costs which, without loss of generality, we set equal to zero. There are no fixed costs. We assume that the technologies of the firms are sufficiently flexible to ensure that there are no constraints on their ability to switch price and output as strategic variables. The market demand curve is Q=q1+q2=cx-Pp. In identifying the demand facing each firm, given the price or output chosen by its rival, we adopt a framework employed by Shubik (1980). Let the aggregated preferences of consumers be represented by a quadratic utility function of the form: 2 aQ_!$-_
2a E
2
I
(1)
i=l
where a-q2
u2=
[
2
y is the measure
2
I
’
of substitutability
of the products
produced
by the duopolists,
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R. Rothschild/J.
of Economic Behavior & Org. 28 (1995) 373-385
and E = 1 + y. We assume that y 2 0, where y = 0 implies that the markets are effectively independent and y = CQimplies that the products are identical ‘. Setting du _=-= a41
au 0 a%
yields the inverse demand curves: 1 a_Q_41 E P [
(2)
=pl
I
and 1
-
a-Q-!EL
P[
(3)
=p2*
E
1
Some further manipulation
yields the demands
for q1 and q2 in terms of p1 and
P21
and
q2=; a+PYPl 2 -4 [
1+;
(5)
p ) 21
The particular strategic framework which we wish to consider here is due to Friedman (1971). Formally, we consider an infinite replication of a single stage game, in which there exists for each player i a pure strategy denoted by the vector zi. At any stage T, TE[O, ~1, Z&T) indicates an action (a choice of price or output) sis, which is a function of the actions of firm j, j # i, in all stages to T - 1, and which yields a payoff rri(si7). For each i let si”?f,si”, and si”, denote the actions corresponding to collusion, Bertrand (or Coumot) behaviour, and deviation from the collusive equilibrium, and let T$s~;), G-~($) and rri( s$> be the associated payoffs. A trigger strategy for the game can then be defined as follows: zi* = s”
zi7 =
s”,
if
$7
otherwise.
zj( x)
=s,F, x=
1, . . . . T-
1;
’ The introduction of product differentiation into the trigger strategy framework [see for instance Deneckere (19831, Ross (1992) and Rothschild (1992)].
is not in itself novel
R. Rothschild/J. of Economic Behavior & Org. 28 (1995) 373-385
377
The essence of Friedman’s analysis is that any collusive equilibrium can be sustained if, for some common discount rate r, the following condition is satisfied:
or, equivalently, for the discount parameter P, /A= l/(1 + r>,
(6)
3. An analysis
of competitive between strategic variables
behaviour
when
firms
may
switch
freely
The basis for the discussion in this section is to be found in the models of Bertrand and Coumot behaviour which are set out, using simplified notation, in the appendix. In these two models, firms are assumed to select a single strategic variable at the commencement of the game, and to adhere to this choice throughout. Thus, if deviation takes place in terms of price (output) then punishment, where it occurs, is also in terms of price (output). The appendix sets out the conditions on p under which deviation from the joint-profit maximising collusive equilibrium will be deterred when both firms employ a single strategic variable. Before proceeding to an analysis of the case in which firms may switch freely between the price and output variables for both the deviation and the punishment phase, it will be useful to consider two simpler cases. In the first, deviation takes the form of a profit-maximising price adjustment, with punishment in terms of an output adjustment by both firms; in the second deviation takes the form of a profit maximising output adjustment, with punishment in terms of a price adjustment by both firms. Using the notation introduced in the appendix, recall that 7rrn= rr: = T;. It then follows that in the first case deterrence requires p > r,,
*Eh - Trm H2 .. = -7rB - Trc’ 9y2+32y+32
= ‘-“ch
(7)
and in the second, p, > r,,
= ““ch 7rc
.’ -
lr;
=
9y2 + 32y+ 32 ’
(8)
Note that r’, > rcs, so that if p is large enough to deter deviation in price when reversion by both firms is to output for the punishment phase, then it will also deter deviation in output when reversion to punishment is in price.
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of Economic Behavior & Org. 28 (1995) 373-385
We turn now to the cases involving free switching between price and output. Consider first the outcomes when the deviant (firm 1) selects output as its strategic variable, while the punisher (firm 2) selects price. Writing (3) in direct form yields K( a - PP,) q2
=
-
741
F
’
Substitution into (2) gives the deviant’s price as a function of its own output and the price and output of the punisher. Multiplying and differentiating the resulting expression by ql, and setting this equal to zero yields the deviant’s profit maximising output, given any p2: 2a+PYP?
The punisher’s profit maximising price, given the deviant’s output, is in turn found by substituting this expression into (3), multiplying by q2, differentiating the resulting expression by p2 and setting this equal to zero. This yields p,=$, where K = 8 + 8y + y2. Substitution, sions yields 416 41
as appropriate,
in the two previous
expres-
+ 207 + 5y2)
=
8K
and LXH q2==.
Further substitution in (2) yields the deviant’s price, pl. The profits of the deviant and the punisher at the ‘output-price’ equilibrium are then: ch _ a2(16 TqP
-
+ 2Oy+ 5~~)~ 16PK2F
(9)
and
%ti-
_-
CY2H2 8pKj7’
(10)
It is straightforward to compare, for both the deviant and the punisher, the profits at this equilibrium with those obtainable at the Bertrand and Coumot equilibria identified in (A.21 and (A.7) respectively. Clearly, while the deviant’s profits in this case can be shown to exceed those of the punisher (for all positive y), for both firms the Coumot equilibrium profits are (again, for all positive y)
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of Economic Behavior & Org. 28 (1995) 373-385
379
greater than the ‘output-price’ equilibrium profits. By contrast, the ‘output-price’ equilibrium profits are greater than the Bertrand equilibrium profits for both firms for all positive y. For purposes of comparison, the profits to the deviant and the punisher at a ‘price-output’ equilibrium (denoted $i and rPq, respectively) are found by simply exchanging their respective profits at the ‘output-price’ equilibrium: ch __nbq
ff2H2
(11)
f3Pf@
and L _ ~r’(16+2Oy+5y~)~ a P4 16j3K2F
.
(12)
It follows that in this case the deviant obtains lower profits, and the punisher higher profits, than each would obtain in the ‘output-price’ equilibrium. The foregoing results can be summarised as follows: For the deviant:
lr;>?rq$>lrP$>7T;.
For the punisher:
?r$Vrp;>~q+r;.
The price and output strategies (denoted p and 4 respectively) open to both the deviant (D) and punisher (L), and the associated payoffs, can be analysed as an extensive form game. An illustration is given in Fig. 1. In this representation the second and third nodes (L and D) are assumed to be arbitrarily close to each other, thus capturing the idea that the deviant can ‘switch’ variable for the punishment phase. We have shown that the choice of output as the strategic variable constitutes for each firm a dominant strategy during the punishment phase. Irrespective of the variable chosen by its rival during the punishment, each firm obtains the highest profits possible by playing output rather than price. Suppose therefore that the Coumot equilibrium (rr; , ?r:) emerges as the strategic choice of both firms for the duration of the punishment. The last stage of the game can be approached from either of two directions: the deviant can elect to deviate in terms of output, in which case the punisher will respond in terms of output, or the deviation can be in terms of price. In the latter event, the punisher will choose to respond in output and the deviant will immediately also switch to output. The natural question is which variable will be selected by the deviant for the purposes of deviation. The answer is found by noting that rib > rrGh. A price deviation yields higher profits for the deviant (and, incidentally, as comparison of (A.4) and (A.91 shows, inflicts larger losses upon its rival) than a quantity deviation for all positive y. Consequently, for this case the condition on p which must be satisfied if deviation is to be deterred is given by r,,.
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Fig. 1. Sustaining
of Economic Behavior & Org. 28 (1995) 373-385
collusion when the choice of strategic variable is endogenous.
The difficulty which arises here is that the selection by the punisher of output in response to any deviation does not represent the most severe punishment which can be inflicted upon the deviant. To see this note that, irrespective of the variable which it selects for the punishment phase, the profits which the deviant obtains when the punisher selects price are lower than those obtained when the punisher selects output. It follows that if p I I”=, the punisher could threaten to select price in order to impose upon the deviant a more severe punishment. If this happens, then the deviant’s best response is, as already noted, to select output. In this case, the game can be characterised by a deviation in terms of price, followed by a punishment in which the punisher selects price while the deviant switches to output. If this occurs, then the condition on p necessary to discourage deviation can be written as
Trihp ’ &q, =
c,, =B
P
m
c,, = -
TqP
K2 F(64 + 80~ + 227~’ + y3) *
(13)
It is clear that an outcome which requires the punisher to select price while the deviant selects output is not subgame pegect, since the punisher could do better by also selecting output. Nevertheless, it is important to identify the most severe punishment that supports cooperation for the highest discount rate [see Abreu
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381
(1986) for a detailed analysis of this and related issues], and to show that this punishment turns out to involve an ‘asymmetric’ equilibrium. In order to render credible the threat to punish in price, the punisher would have to commit in some way to the strategy. The question of the nature of such commitment, and the means by which it might be achieved is, however, beyond the scope of the present discussion, as too is a more comprehensive treatment of ‘optimal’ supergame strategies in general [see Abreu (1986)]. An important insight to emerge from the foregoing analysis is that since fBc > I&, > fBB > I& > Fcs, it follows that when deviation takes the form of a price adjustment, the magnitude of I_Lnecessary to deter deviation is greater than that necessary when deviation is in output. Moreover, whatever the strategic variable selected for the deviation, such deviation is more easily deterred (that is, the required p is smaller) when the deviant selects price for the punishment phase. The intuition behind these results is that (a) the greater the degree of product substitutability (that is, the larger is y) the greater the ‘damage’ which a given deviation and punishment will inflict upon the rival firm, and (b) given any y, the ‘damage’ inflicted by such deviation (punishment) will be greater if price is the selected strategic variable. The reason for the first observation is self-evident: the lower is y, the more insulated is each firm’s market from that of its rival. The second observation derives from the well-known result that, other things equal, the Bertrand equilibrium is identified with a lower price and profit for each firm than the Coumot equilibrium. 4. Concluding comments The purpose of this paper has been to bring to bear on the concept of a ‘trigger strategy’ a question which has been raised in the related context of the comparative statics literature by authors such as Hathaway and Rickard (19791, Singh and Vives (1984) and Cheng (1985). These authors have compared the outcomes, in terms of price and output, inter alia, of switching by firms between Bertrand and Coumot behaviour. In this paper we employ a linear demand framework similar to that used by earlier authors, in order to show the roles of the discount parameter and the degree of product substitutability in determining the extent to which collusive agreements in repeated games may be robust against the individual incentive to deviate when switching by both the prospective deviant and the punisher is permitted. In particular we have sought to answer two related questions. The first concerns the smallest value of the discount parameter p which will deter deviation when firms may choose freely between the price and output variables. The second concerns the possibility of an asymmetrical outcome to the implementation of the punishment strategy. As Klemperer and Meyer have noted for the ‘one-shot’ game, there could exist in addition to the Bertrand and Coumot equilibria, two asymmetric equilibria in which one firm sets price and the other sets output.
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of Economic Behavior & Org. 28 (1995) 373-385
The analysis offered here goes some way towards resolving these questions in the context of a supergame. In this framework the convention is to assume, without formal justification, that the strategic variables chosen by both the deviant and punisher will coincide, so that if deviation takes place in terms of price (output), then punishment, where it occurs, will similarly be in terms of price (output). We have shown that where switching in variables is permitted, this assumption may not be valid, and that in the linear framework at least deviation would, from the deviant’s point of view, ideally be in terms of price with punishment in terms of output. This combination is subgameperfect because the selection of output in the punishment phase is a dominant strategy for both firms. By contrast, the ‘most severe’ punishment would involve an asymmetry in the selection of strategic variables, with the punisher selecting price and the deviant switching to output. Such an outcome is, however, not subgame perfect, and would have to be supported by some kind of commitment on the part of the punisher. The possibility that the punisher might wish to employ a ‘most severe’ punishment rather than a reversion to noncooperation raises questions of the kind considered by Abreu (1986) in the context of ‘optimal’ supergame strategies. As we have already noted, however, these issues lie outside of the scope of the present discussion. Nevertheless, it would clearly be interesting to study in more depth than has been possible here the relationship between the particular choice of deviation and punishment strategy, on the one hand, and the degree to which collusion can be sustained, on the other.
Acknowledgements The author wishes to thank two anonymous comments on an earlier draft.
referees
for numerous
helpful
Appendix A Bertrand behaviour If firms compete in price then the demand for the output of firms 1 and 2 is given by the expressions in (4) and (5) respectively. By contrast, at the joint profit maximum, p = a/2P and each firm sells the quantity a/4 to obtain 2 TB
m=-
cl2
SP’
* In what follows we shall employ a slightly simpler notation than heretofore, setting mi(sE) = mm, T~(s:,)= vc and ~~(s~$‘h)= rCk. We shall distinguish, where appropriate, between Bertrand and Cournot behaviour by means of the subscripts B and C respectively. The term ?yL will be defined separately.
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The condition
ofEconomic Behavior & Org. 28 (1995) 373-385
for a Bertrand noncooperative
equilibrium
383
is
(A.11 for i = 1, 2; j = 2, 1. Solving yields pi = (~/@(2
qi = aF/2G
and
a2(l + Y/2)
~TTc = B
+ Y/2),
2P(2
(A.4
’
+ Y/2)2
where F=2+ y and G=4+ y. We use the expression in (A.11 to find the profit maximising price deviation for firm i, given the loyalist’s price pj = a/2P, and then substitute the two prices in (4) to obtain the deviant’s output (in this case, ql). This procedure yields p1 = CYG/~PF, q, = aG/16 and ch __TB
Ct2G2
(A-3)
64PF ’
If a profit maximising deviation obtained by the loyalist are L_ TB
of this kind occurs, then the ‘one-period’
_ a2(Y2-4Y-8).
-
profits
(A-4)
32/3F
Note that TT~ I zk, so that the threat of reversion to noncooperative behaviour credible in the sense that it is more profitable for the punisher to carry out threat than it is to let the deviant go unpunished. The condition on the common discount parameter p which must be satisfied deviation is to be deterred is found by substituting T;, 7rih and T; into (6) obtain IrB
CL>
ch
-
r,, = 7rih-
(A4
= y2+16y+32’
Cournot behauiour If firms compete in output, then the profits to each at the joint-profit are as before:
Again following equilibrium is
--a a4i
4i
[P
Shubik
if to
G2
Tr; T;
is its
(1980),
the condition
(a_Q_4i-4’ )I
=o,
E
for a Coumot
maximum
noncooperative
(i-9
384
R. Rothschild/J.
for i = 1,2; j = 2,l. Solving and
ofEconomic Behavior & Org. 28 (1995) 373-385
yields
qi =
aE/H,
a2EF = pH2
+
pi = cr(l + y/2)/P@
+ 3-y/2)
(A.7)
where H = 4 + 3~. In order to find the profit-maximising output deviation for firm i, we set the output of the loyalist, qj, equal to (-w/4. Substitution of this value of qj in (A.6) then yields a profit-maximising qi = aH/8F. Substitution in (2) yields pi = aH/8PE and a profit for the deviant of (u2H2
ch __ TC
(A-8)
64pFE
Since there is an asymmetry in the outputs of the two firms there is also an asymmetry in prices. In this case the price at which the loyalist sells its output of (~/4 is found by substituting qi and qj in (3) to obtain ffM SPFE
‘j=
and its ‘one-period’ &
-
~
profit is
ff2M
’ - 32fIFE’
(A.91
where M = 8 + 12-y + 3~‘. Note that in this case, too, ri I rrd, so that the threat of noncooperation is again credible in the sense that the loyal firm finds implementation profitable. The condition necessary to deter deviation is found by substituting as appropriate in (6):
H2 7Tg”- lr; CL’ rCC= lTc ch - 7T; = 17y2+48y+32’
(A.lO)
It is important to note that Z& < r’,. It follows that the value of the discount parameter p which sustains collusion in an exclusively price adjusting cartel will also sustain collusion when firms adjust outputs exclusively, but not vice-versa.
References Abreu, D., 1986, Extremal equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191-225. Al-Nowaihi, A. and P. Levine, 1992, Credibility and the degree of collusion among oligopolists, University of L&ester (mimeo).
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of Economic Behavior & Org. 28 (1995) 373-385
385
Cheng, L., 1985, Comparing Bertrand and Cournot equilibria: a geometric approach, Rand Journal of Economics 16, 146-152. Deneckere, R., 1983, Duopoly supergames with product differentiation, Economics Letters 11, 37-42. Friedman, J., 1971, A noncooperative equilibrium for supergames, Review of Economic Studies 38, l-12. Hathaway, N.J. and J.A. Rickard, 1979, Equilibria of price-setting and quantity-setting duopolies, Economics Letters 3, 133-137. Klemperer, P. and M. Meyer, 1986, Price competition vs quantity competition: the role of uncertainty, Rand Journal of Economics 17, 618-638. Ross, T.W., 1992, Cartel stability and product differentiation. International Journal of Industrial Organization 10, l-13. Rothschild, R., 1985, Noncooperative behaviour as a credible threat, Bulletin of Economic Research 37, 245-248. Rothschild, R., 1992, On the sustainability of collusion in differentiated duopolies, Economics Letters 40, 33-37. Shubik, M., 1980, Market structure and behavior, Harvard University Press, Cambridge Mass. Singh, N. and X. Vives, 1984, Price and quantity competition in a differentiated duopoly, Rand Journal of Economics 15. 546-554.
Journal of Economic Behavior and Organization Vol. 28 (1995) 373-385
EconomicBehavior & Oganization
Sustaining collusion when the choice of strategic variable is endogenous R. Rothschild Lancaster University, Department of Economics, Lancaster LA1 4Ex, UK
Received 14 February 1994; revised 29 August 1994
Abstract This paper addresses the problem of maintaining collusion in a differentiated duopoly when both the prospective deviant and punisher are free to choose between price and output as their strategic variables. We show that the traditional formulation of the supergame in which fiis commit for the duration to either price or output as their single common strategic variable overlooks the possibility of an asymmetric equilibrium in the ‘punishment’ phase. The results obtained here have obvious parallels with those found in the comparative statics literature. JEL classification: Keywords:
D43
Collusion; Supergame; Punishment; Price and output strategies
1. Introduction
In recent years the literature of oligopoly theory has been dominated by the search for mechanisms which will maintain the integrity of collusive arrangements in the face of the potential for deviant behaviour. The best known examples are due to Friedman (1971) and Abreu (1986). The underlying principle is that firms agree to adhere to a particular price or output level for as long as their fellow oligopolists do the same, but either revert to noncooperative behaviour forever in the event of deviation from this agreement (Friedman), or adopt a ‘carrot and stick’ rule which encourages even more severe punishments of finite duration followed by a return to cooperation (Abreu). In these models if, for any prospec0167-2681/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2681(95)00041-O
374
R. Rothschild/J.
of Economic Behavior & Org. 28 (1995) 373-385
tive deviant, the one-period gain from deviation is less than the (discounted) value of the losses arising from subsequent punishment, the cartel price or output will be maintained. Although the credibility of the threat which under-pins such strategies has been the subject of some discussion [see, for example, Rothschild (1985) and Al-Nowaihi and Levine (199211 the concept of the ‘trigger strategy’ has now become widely accepted. In light of this fact it seems appropriate to consider a neglected issue. This concerns the implicit assumption in the earlier literature that if price (output) is selected initially by the coalition as the strategic variable, then any deviation were it to occur - and any subsequent ‘punishment’ will be implemented in terms of that variable. In other words, the threat strategy requires that there be no switching by any firm between the two strategic variables once the initial choice has been made. The requirement that the variable be exogenously given in this way seems both unreasonable and restrictive, especially when one considers markets in which production capacity is highly flexible. For example, in their analysis of a related problem, Klemperer and Meyer (1986) consider the case of a consultant who chooses between a fee per hour and a fixed fee for a project as a whole. By charging per hour the consultant is setting a price; in charging for the project as a whole he or she has the option of spending less time on the client when the office is busy than when it is not (so that the work day is constant) and is in this sense setting a quantity and adjusting price to demand. More familiar examples of this phenomenon can be found in commercial aviation and in the oil industry, in which firms may compete either in price or in output. For instance, it is well known that airline operators have the option of either cutting price per seat on a given route or increasing the number of seats on offer at the prevailing price. Similarly, oil producers may elect to either undercut their rivals in price or ‘cheat’ on an agreement by increasing their output beyond quota. In markets of this type sellers can, moreover, switch easily between the price and output variables, and it is difficult to conceive of any mechanism which will prevent such switching if it is in the interests of agents to do so. In this paper we consider possible outcomes when the deviant is able to select a strategic variable different from that chosen initially by the coalition, and the ‘loyal’ member, given this selection, is similarly in a position to choose for the punishment phase the variable which yields to it the highest profits available. In focusing on the problem in the context of a supergame this paper goes beyond a related literature which has thus far been restricted to the comparison of Bertrand and Cournot behaviour [see, for example. Hathaway and Rickard (1979), Singh and Vives (19841, and Cheng (1985)]. The emphasis in these latter contributions has been largely on comparing the associated equilibria of the models (in terms of price, output and welfare) in a static framework. Once the constraints imposed by this framework are relaxed, it becomes possible to consider explicitly the implications of ‘switching’. In particular, we wish in this paper to consider the possibility that (a) the firm
R. Rothschild/J.
ofEconomic Behavior & Org. 28 (1995) 373-385
375
which undertakes the punishment will be able to choose between the price and output variables for the ‘punishment’ phase, in order if necessary to threaten the most effective response (in terms of the power to deter) to deviant behaviour, and (b> the deviant will select for itself both the most profitable form of deviation and the most appropriate response to the punishment variable which the punisher threatens to choose. In addressing these questions we shall show, inter alia, that when firms are able to choose freely between the price and output variables, the preferred choice of the deviant will be price for the initial deviation, followed by output for the punishment. By contrast, however, we shall show also that while subgame perfection may dictate that in this case the punisher also chooses output for the punishment phase, the prospect of a ‘more severe’ punishment involving an asymmetric equilibrium might constitute a more effective deterrent to deviation. The paper is organised as follows. Section 2 sets out the model. Section 3 considers the implications of permitting firms to choose freely between the price and output variables. Section 4 offers some concluding comments. An Appendix sets out the two ‘benchmark’ cases of Bertrand and Cournot competition which will serve as a basis for the discussion in Section 3. Some simplifying notation is also introduced.
2. The model We consider a duopoly in which both firms i,i = 1,2, produce their output qi subject to identical, constant marginal costs which, without loss of generality, we set equal to zero. There are no fixed costs. We assume that the technologies of the firms are sufficiently flexible to ensure that there are no constraints on their ability to switch price and output as strategic variables. The market demand curve is Q=q1+q2=cx-Pp. In identifying the demand facing each firm, given the price or output chosen by its rival, we adopt a framework employed by Shubik (1980). Let the aggregated preferences of consumers be represented by a quadratic utility function of the form: 2 aQ_!$-_
2a E
2
I
(1)
i=l
where a-q2
u2=
[
2
y is the measure
2
I
’
of substitutability
of the products
produced
by the duopolists,
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of Economic Behavior & Org. 28 (1995) 373-385
and E = 1 + y. We assume that y 2 0, where y = 0 implies that the markets are effectively independent and y = CQimplies that the products are identical ‘. Setting du _=-= a41
au 0 a%
yields the inverse demand curves: 1 a_Q_41 E P [
(2)
=pl
I
and 1
-
a-Q-!EL
P[
(3)
=p2*
E
1
Some further manipulation
yields the demands
for q1 and q2 in terms of p1 and
P21
and
q2=; a+PYPl 2 -4 [
1+;
(5)
p ) 21
The particular strategic framework which we wish to consider here is due to Friedman (1971). Formally, we consider an infinite replication of a single stage game, in which there exists for each player i a pure strategy denoted by the vector zi. At any stage T, TE[O, ~1, Z&T) indicates an action (a choice of price or output) sis, which is a function of the actions of firm j, j # i, in all stages to T - 1, and which yields a payoff rri(si7). For each i let si”?f,si”, and si”, denote the actions corresponding to collusion, Bertrand (or Coumot) behaviour, and deviation from the collusive equilibrium, and let T$s~;), G-~($) and rri( s$> be the associated payoffs. A trigger strategy for the game can then be defined as follows: zi* = s”
zi7 =
s”,
if
$7
otherwise.
zj( x)
=s,F, x=
1, . . . . T-
1;
’ The introduction of product differentiation into the trigger strategy framework [see for instance Deneckere (19831, Ross (1992) and Rothschild (1992)].
is not in itself novel
R. Rothschild/J. of Economic Behavior & Org. 28 (1995) 373-385
377
The essence of Friedman’s analysis is that any collusive equilibrium can be sustained if, for some common discount rate r, the following condition is satisfied:
or, equivalently, for the discount parameter P, /A= l/(1 + r>,
(6)
3. An analysis
of competitive between strategic variables
behaviour
when
firms
may
switch
freely
The basis for the discussion in this section is to be found in the models of Bertrand and Coumot behaviour which are set out, using simplified notation, in the appendix. In these two models, firms are assumed to select a single strategic variable at the commencement of the game, and to adhere to this choice throughout. Thus, if deviation takes place in terms of price (output) then punishment, where it occurs, is also in terms of price (output). The appendix sets out the conditions on p under which deviation from the joint-profit maximising collusive equilibrium will be deterred when both firms employ a single strategic variable. Before proceeding to an analysis of the case in which firms may switch freely between the price and output variables for both the deviation and the punishment phase, it will be useful to consider two simpler cases. In the first, deviation takes the form of a profit-maximising price adjustment, with punishment in terms of an output adjustment by both firms; in the second deviation takes the form of a profit maximising output adjustment, with punishment in terms of a price adjustment by both firms. Using the notation introduced in the appendix, recall that 7rrn= rr: = T;. It then follows that in the first case deterrence requires p > r,,
*Eh - Trm H2 .. = -7rB - Trc’ 9y2+32y+32
= ‘-“ch
(7)
and in the second, p, > r,,
= ““ch 7rc
.’ -
lr;
=
9y2 + 32y+ 32 ’
(8)
Note that r’, > rcs, so that if p is large enough to deter deviation in price when reversion by both firms is to output for the punishment phase, then it will also deter deviation in output when reversion to punishment is in price.
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of Economic Behavior & Org. 28 (1995) 373-385
We turn now to the cases involving free switching between price and output. Consider first the outcomes when the deviant (firm 1) selects output as its strategic variable, while the punisher (firm 2) selects price. Writing (3) in direct form yields K( a - PP,) q2
=
-
741
F
’
Substitution into (2) gives the deviant’s price as a function of its own output and the price and output of the punisher. Multiplying and differentiating the resulting expression by ql, and setting this equal to zero yields the deviant’s profit maximising output, given any p2: 2a+PYP?
The punisher’s profit maximising price, given the deviant’s output, is in turn found by substituting this expression into (3), multiplying by q2, differentiating the resulting expression by p2 and setting this equal to zero. This yields p,=$, where K = 8 + 8y + y2. Substitution, sions yields 416 41
as appropriate,
in the two previous
expres-
+ 207 + 5y2)
=
8K
and LXH q2==.
Further substitution in (2) yields the deviant’s price, pl. The profits of the deviant and the punisher at the ‘output-price’ equilibrium are then: ch _ a2(16 TqP
-
+ 2Oy+ 5~~)~ 16PK2F
(9)
and
%ti-
_-
CY2H2 8pKj7’
(10)
It is straightforward to compare, for both the deviant and the punisher, the profits at this equilibrium with those obtainable at the Bertrand and Coumot equilibria identified in (A.21 and (A.7) respectively. Clearly, while the deviant’s profits in this case can be shown to exceed those of the punisher (for all positive y), for both firms the Coumot equilibrium profits are (again, for all positive y)
R. Rothschild/J.
of Economic Behavior & Org. 28 (1995) 373-385
379
greater than the ‘output-price’ equilibrium profits. By contrast, the ‘output-price’ equilibrium profits are greater than the Bertrand equilibrium profits for both firms for all positive y. For purposes of comparison, the profits to the deviant and the punisher at a ‘price-output’ equilibrium (denoted $i and rPq, respectively) are found by simply exchanging their respective profits at the ‘output-price’ equilibrium: ch __nbq
ff2H2
(11)
f3Pf@
and L _ ~r’(16+2Oy+5y~)~ a P4 16j3K2F
.
(12)
It follows that in this case the deviant obtains lower profits, and the punisher higher profits, than each would obtain in the ‘output-price’ equilibrium. The foregoing results can be summarised as follows: For the deviant:
lr;>?rq$>lrP$>7T;.
For the punisher:
?r$Vrp;>~q+r;.
The price and output strategies (denoted p and 4 respectively) open to both the deviant (D) and punisher (L), and the associated payoffs, can be analysed as an extensive form game. An illustration is given in Fig. 1. In this representation the second and third nodes (L and D) are assumed to be arbitrarily close to each other, thus capturing the idea that the deviant can ‘switch’ variable for the punishment phase. We have shown that the choice of output as the strategic variable constitutes for each firm a dominant strategy during the punishment phase. Irrespective of the variable chosen by its rival during the punishment, each firm obtains the highest profits possible by playing output rather than price. Suppose therefore that the Coumot equilibrium (rr; , ?r:) emerges as the strategic choice of both firms for the duration of the punishment. The last stage of the game can be approached from either of two directions: the deviant can elect to deviate in terms of output, in which case the punisher will respond in terms of output, or the deviation can be in terms of price. In the latter event, the punisher will choose to respond in output and the deviant will immediately also switch to output. The natural question is which variable will be selected by the deviant for the purposes of deviation. The answer is found by noting that rib > rrGh. A price deviation yields higher profits for the deviant (and, incidentally, as comparison of (A.4) and (A.91 shows, inflicts larger losses upon its rival) than a quantity deviation for all positive y. Consequently, for this case the condition on p which must be satisfied if deviation is to be deterred is given by r,,.
380
R. Rothschild/J.
Fig. 1. Sustaining
of Economic Behavior & Org. 28 (1995) 373-385
collusion when the choice of strategic variable is endogenous.
The difficulty which arises here is that the selection by the punisher of output in response to any deviation does not represent the most severe punishment which can be inflicted upon the deviant. To see this note that, irrespective of the variable which it selects for the punishment phase, the profits which the deviant obtains when the punisher selects price are lower than those obtained when the punisher selects output. It follows that if p I I”=, the punisher could threaten to select price in order to impose upon the deviant a more severe punishment. If this happens, then the deviant’s best response is, as already noted, to select output. In this case, the game can be characterised by a deviation in terms of price, followed by a punishment in which the punisher selects price while the deviant switches to output. If this occurs, then the condition on p necessary to discourage deviation can be written as
Trihp ’ &q, =
c,, =B
P
m
c,, = -
TqP
K2 F(64 + 80~ + 227~’ + y3) *
(13)
It is clear that an outcome which requires the punisher to select price while the deviant selects output is not subgame pegect, since the punisher could do better by also selecting output. Nevertheless, it is important to identify the most severe punishment that supports cooperation for the highest discount rate [see Abreu
R. Rothschild/J. of Economic Behavior & Org. 28 (1995) 373-385
381
(1986) for a detailed analysis of this and related issues], and to show that this punishment turns out to involve an ‘asymmetric’ equilibrium. In order to render credible the threat to punish in price, the punisher would have to commit in some way to the strategy. The question of the nature of such commitment, and the means by which it might be achieved is, however, beyond the scope of the present discussion, as too is a more comprehensive treatment of ‘optimal’ supergame strategies in general [see Abreu (1986)]. An important insight to emerge from the foregoing analysis is that since fBc > I&, > fBB > I& > Fcs, it follows that when deviation takes the form of a price adjustment, the magnitude of I_Lnecessary to deter deviation is greater than that necessary when deviation is in output. Moreover, whatever the strategic variable selected for the deviation, such deviation is more easily deterred (that is, the required p is smaller) when the deviant selects price for the punishment phase. The intuition behind these results is that (a) the greater the degree of product substitutability (that is, the larger is y) the greater the ‘damage’ which a given deviation and punishment will inflict upon the rival firm, and (b) given any y, the ‘damage’ inflicted by such deviation (punishment) will be greater if price is the selected strategic variable. The reason for the first observation is self-evident: the lower is y, the more insulated is each firm’s market from that of its rival. The second observation derives from the well-known result that, other things equal, the Bertrand equilibrium is identified with a lower price and profit for each firm than the Coumot equilibrium. 4. Concluding comments The purpose of this paper has been to bring to bear on the concept of a ‘trigger strategy’ a question which has been raised in the related context of the comparative statics literature by authors such as Hathaway and Rickard (19791, Singh and Vives (1984) and Cheng (1985). These authors have compared the outcomes, in terms of price and output, inter alia, of switching by firms between Bertrand and Coumot behaviour. In this paper we employ a linear demand framework similar to that used by earlier authors, in order to show the roles of the discount parameter and the degree of product substitutability in determining the extent to which collusive agreements in repeated games may be robust against the individual incentive to deviate when switching by both the prospective deviant and the punisher is permitted. In particular we have sought to answer two related questions. The first concerns the smallest value of the discount parameter p which will deter deviation when firms may choose freely between the price and output variables. The second concerns the possibility of an asymmetrical outcome to the implementation of the punishment strategy. As Klemperer and Meyer have noted for the ‘one-shot’ game, there could exist in addition to the Bertrand and Coumot equilibria, two asymmetric equilibria in which one firm sets price and the other sets output.
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of Economic Behavior & Org. 28 (1995) 373-385
The analysis offered here goes some way towards resolving these questions in the context of a supergame. In this framework the convention is to assume, without formal justification, that the strategic variables chosen by both the deviant and punisher will coincide, so that if deviation takes place in terms of price (output), then punishment, where it occurs, will similarly be in terms of price (output). We have shown that where switching in variables is permitted, this assumption may not be valid, and that in the linear framework at least deviation would, from the deviant’s point of view, ideally be in terms of price with punishment in terms of output. This combination is subgameperfect because the selection of output in the punishment phase is a dominant strategy for both firms. By contrast, the ‘most severe’ punishment would involve an asymmetry in the selection of strategic variables, with the punisher selecting price and the deviant switching to output. Such an outcome is, however, not subgame perfect, and would have to be supported by some kind of commitment on the part of the punisher. The possibility that the punisher might wish to employ a ‘most severe’ punishment rather than a reversion to noncooperation raises questions of the kind considered by Abreu (1986) in the context of ‘optimal’ supergame strategies. As we have already noted, however, these issues lie outside of the scope of the present discussion. Nevertheless, it would clearly be interesting to study in more depth than has been possible here the relationship between the particular choice of deviation and punishment strategy, on the one hand, and the degree to which collusion can be sustained, on the other.
Acknowledgements The author wishes to thank two anonymous comments on an earlier draft.
referees
for numerous
helpful
Appendix A Bertrand behaviour If firms compete in price then the demand for the output of firms 1 and 2 is given by the expressions in (4) and (5) respectively. By contrast, at the joint profit maximum, p = a/2P and each firm sells the quantity a/4 to obtain 2 TB
m=-
cl2
SP’
* In what follows we shall employ a slightly simpler notation than heretofore, setting mi(sE) = mm, T~(s:,)= vc and ~~(s~$‘h)= rCk. We shall distinguish, where appropriate, between Bertrand and Cournot behaviour by means of the subscripts B and C respectively. The term ?yL will be defined separately.
R. Rothschild/J.
The condition
ofEconomic Behavior & Org. 28 (1995) 373-385
for a Bertrand noncooperative
equilibrium
383
is
(A.11 for i = 1, 2; j = 2, 1. Solving yields pi = (~/@(2
qi = aF/2G
and
a2(l + Y/2)
~TTc = B
+ Y/2),
2P(2
(A.4
’
+ Y/2)2
where F=2+ y and G=4+ y. We use the expression in (A.11 to find the profit maximising price deviation for firm i, given the loyalist’s price pj = a/2P, and then substitute the two prices in (4) to obtain the deviant’s output (in this case, ql). This procedure yields p1 = CYG/~PF, q, = aG/16 and ch __TB
Ct2G2
(A-3)
64PF ’
If a profit maximising deviation obtained by the loyalist are L_ TB
of this kind occurs, then the ‘one-period’
_ a2(Y2-4Y-8).
-
profits
(A-4)
32/3F
Note that TT~ I zk, so that the threat of reversion to noncooperative behaviour credible in the sense that it is more profitable for the punisher to carry out threat than it is to let the deviant go unpunished. The condition on the common discount parameter p which must be satisfied deviation is to be deterred is found by substituting T;, 7rih and T; into (6) obtain IrB
CL>
ch
-
r,, = 7rih-
(A4
= y2+16y+32’
Cournot behauiour If firms compete in output, then the profits to each at the joint-profit are as before:
Again following equilibrium is
--a a4i
4i
[P
Shubik
if to
G2
Tr; T;
is its
(1980),
the condition
(a_Q_4i-4’ )I
=o,
E
for a Coumot
maximum
noncooperative
(i-9
384
R. Rothschild/J.
for i = 1,2; j = 2,l. Solving and
ofEconomic Behavior & Org. 28 (1995) 373-385
yields
qi =
aE/H,
a2EF = pH2
+
pi = cr(l + y/2)/P@
+ 3-y/2)
(A.7)
where H = 4 + 3~. In order to find the profit-maximising output deviation for firm i, we set the output of the loyalist, qj, equal to (-w/4. Substitution of this value of qj in (A.6) then yields a profit-maximising qi = aH/8F. Substitution in (2) yields pi = aH/8PE and a profit for the deviant of (u2H2
ch __ TC
(A-8)
64pFE
Since there is an asymmetry in the outputs of the two firms there is also an asymmetry in prices. In this case the price at which the loyalist sells its output of (~/4 is found by substituting qi and qj in (3) to obtain ffM SPFE
‘j=
and its ‘one-period’ &
-
~
profit is
ff2M
’ - 32fIFE’
(A.91
where M = 8 + 12-y + 3~‘. Note that in this case, too, ri I rrd, so that the threat of noncooperation is again credible in the sense that the loyal firm finds implementation profitable. The condition necessary to deter deviation is found by substituting as appropriate in (6):
H2 7Tg”- lr; CL’ rCC= lTc ch - 7T; = 17y2+48y+32’
(A.lO)
It is important to note that Z& < r’,. It follows that the value of the discount parameter p which sustains collusion in an exclusively price adjusting cartel will also sustain collusion when firms adjust outputs exclusively, but not vice-versa.
References Abreu, D., 1986, Extremal equilibria of oligopolistic supergames, Journal of Economic Theory 39, 191-225. Al-Nowaihi, A. and P. Levine, 1992, Credibility and the degree of collusion among oligopolists, University of L&ester (mimeo).
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of Economic Behavior & Org. 28 (1995) 373-385
385
Cheng, L., 1985, Comparing Bertrand and Cournot equilibria: a geometric approach, Rand Journal of Economics 16, 146-152. Deneckere, R., 1983, Duopoly supergames with product differentiation, Economics Letters 11, 37-42. Friedman, J., 1971, A noncooperative equilibrium for supergames, Review of Economic Studies 38, l-12. Hathaway, N.J. and J.A. Rickard, 1979, Equilibria of price-setting and quantity-setting duopolies, Economics Letters 3, 133-137. Klemperer, P. and M. Meyer, 1986, Price competition vs quantity competition: the role of uncertainty, Rand Journal of Economics 17, 618-638. Ross, T.W., 1992, Cartel stability and product differentiation. International Journal of Industrial Organization 10, l-13. Rothschild, R., 1985, Noncooperative behaviour as a credible threat, Bulletin of Economic Research 37, 245-248. Rothschild, R., 1992, On the sustainability of collusion in differentiated duopolies, Economics Letters 40, 33-37. Shubik, M., 1980, Market structure and behavior, Harvard University Press, Cambridge Mass. Singh, N. and X. Vives, 1984, Price and quantity competition in a differentiated duopoly, Rand Journal of Economics 15. 546-554.