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Model consistent beliefs with information transmission in oligopoly
European Journal of Political Economy 8 (1992) 231-249. North-Holland
Model consistent beliefs with information transmission in oligopoly* Morten
Hviid and Norman
Department
of Economics, University
J. Ireland
of Warwick,
Coventry,
CV4
7AL,
UK
Accepted for publication November 1991
A model of market coordination in the presence of private information is developed. One component is a period of talk about output levels. The talk ends when no firm wishes to change its latest announced planned output level, and only if all firms enact their latest announced plans will such talk be repeated in future periods. It is shown how the degree of coordination and apparent collusion achieved is related to the noise in signalling market information.
1. Introduction
The ability of firms in an oligopolistic industry to reach agreements to restrict production levels and enhance profits is based on two key factors. First, it must be possible to observe (at least to some extent) whether or not individual firms cheat on their agreements. Second, a disciplinary device must be available to punish cheating, so that the agreements are incentive compatible. From the folk theorems for repeated games we know that such discipline is available through an infinite repetition of the stage game, so that short-term profits from cheating are outweighed by punishment in later periods. The problem of non-observability of firms’ actions limits the extent of collusive agreements.’ This complication is especially limiting if uncertainty is (partly) resolved before firms choose their individual strategy. An example of this is discussed in Rotemberg and Saloner (1986).’ A related literature [founded by Frisch (1933)] focusses on the concept of lHviid gratefully acknowledges financial support from the Danish Social Science Research Council. Preliminary versions of this paper have been presented at seminars in the Universities of Aarhus, Copenhagen, Sheffield and Warwick, and at Birkbeck College, London. We are grateful to participants of these seminars for many helpful suggestions and comments. Particular thanks are due to Torben Andersen, Paul Geroski, Xavier Vives, and anonymous referees of this journal. ‘See for instance Green and Porter (1984). ‘In their model, lirms know the current realisation of the random variable before they choose their current strategy, i.e., whether or not to comply with the collusive arrangement. The collusive arrangement, on the other hand, is made before the realisation and is based on expected values. Similarly the consequence of a punishment phase is based on expected value. Thus there exist sufficiently good realisations for deviations to be profitable. 0176-2680/92/505.00 0 1992-Elsevier
Science Publishers B.V. All rights reserved
232
,M. Hciid and N.J.
Ireland,
Model
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beliefs wirh information
transmission
in oligopoly
output response conjectures. This literature has attempted to determine the degree of apparent collusion in an industry by asking what conjectured output reponses are consistent with actual responses [see for instance Boyer and Moreaux (1983a, b), Bresnahan (1981), Perry (1982) and Ulph (1983)]. The outcome has been disappointing. Either too limited a set of conjectural equilibria have been found to account for observed variations, or it has been claimed that no refinement can be achieved, and that any equilibrium is a consistent conjectures equilibrium for some local conjectural function [Boyer and Moreaux (1983a)]. This literature has not encompassed questions of information and observability; nor has it in general incorporated dynamic aspects of repeated interaction. 3 It seems necessary therefore to shift ‘the emphasis . . . from the equilibrium concept itself to the origins, development and validation of conjectures’ [Bayer and Moreaux (1983, p. 40)]. In this paper we go some way towards offering an endogenous explanation of conjectures in terms of the ability of firms to signal market information. Further, the level of current period coordination is supported by expected profits in future periods. We focus on coordination within a market which is affected by stochastic shocks in each period. Then market shares or output quotas cannot be allocated without common agreement concerning the size of the market. We would naturally expect each firm to have its own view as to this size, perhaps formed from its own private information obtained from its market contacts. We will argue that the pooling of such private information is fraught with ditliculties. In particular, if firms are heterogeneous with respect to their costs of production (and particularly if a firm’s cost level is also private information) then firms will have an incentive to provide an information input to the industry cartel which is biased by their own cost positions. If a firm anticipates higher costs than faced by other firms (perhaps because of employees’ bargaining power) then it might claim that the market will be very small, thus trying to reduce all firms’ output quotas and change the balance between high price and sales volume to favour its own cost position. Relying on individual firms to supply information which has no public observable source and is thus unveritiable is not satisfactory. The problem is that talk is ‘cheap’; the firm does not have to behave in any way consistent with the information it supplies. It does not necessarily reveal either its true costs or its true initial market information. Furthermore the level of punishments is more difficult to judge as the extent of deviations from truth telling cannot be observed. As an alternative to such cheap talk, consider the following. Firms may submit output plans which can be modified after listening to other firms output plans. Imagine a blackboard where each firm’s initial plan (a size of 3An exception
to the latter is Kalai
and Stanford
(1985).
M. Hoiid and N.J. Ireland,
Model
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beliefs with information
fransmission
in oligopoly
233
output) is chalked up. Then any firm can erase and revise its plan. When no firm wishes to revise its plan further, given all other firms’ plans, these are the outputs adopted by the industry. Actual outputs are observable and any deviation from Sinai planned outputs are punished in later periods. Basically each firm is forced to put its money where its mouth is. In a full information version of the above game many outcomes may be supportable as perfect equilibria depending on an appropriate disciplining device in the overafl supergame. In a model with imperfect info~ation, the concept of the ‘right’ industry output or the ‘right’ output level for a particular firm is not well defined, since information transmission wilt always be manipulated by firms for their own private gains. However, it is still possible to discipline firms to reveal their final output intentions, and then to implement them, provided that the game played ‘with the blackboard’ yields more expected payoff than that without such coordinating interaction. Basically, true output intention is a verifiable and significant part of the firm’s behaviour. It contains information. Not only is it a way of committing to supply the market (in the Cournot-Nash sense) but it is also a way of indicating a view of the potential market size. No firm wishes to give the impression it believes that demand is high since this would encourage high output decisions from other firms. On the other hand low output plans imply that more optimistic firms would average the pessimistic signal with their own information and tend to achieve a higher market share. Thus when a holiday tour operator states that it plans to sell, say, f million holidays in the next year, this is partly market preemption and partly an indication of that firm’s expectation of the market size next year. The latter information can be at least partially inferred since output plans which are to be implemented are not ‘cheap talk’. They involve actions which involve profits or losses. That ‘tinal’ plans have to be implemented means that, although the planned market supplies can be changed in response to other firms supply plans, they cannot be changed secretly, nor can they be changed after other firms plans are committed. In the holiday tour industry, secrecy is unlikely. More importantly the number of holidays provided becomes public information. Thus cheating will be found out and will lead to coordination breaking down in future periods. This is a disincentive to cheat provided that the blackboard game yields on average a reduction in total supply and more monopoly pro&s. The information signalhng aspect of the blackboard game depends on how far the market conditions observed by one firm are relevant for the profitability of others. If each tour operator sold a rather different type of holiday to other operators, there would be less to be learned. We model this ‘noise’ as firms differing with respect to an independent random variable on their costs, but this is just for clarity. The key point is that, as the noise becomes more significant, the information motive beomes less important and
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hf. Hviid and N.J. Ireland, Model consisrent beliefs with information
transmission
in oligopoly
each firm simply tries to preempt as much of the market as possible, leading to a more competitive outcome. In this way the degree of monopoly observed in an industry may be determined by the relative importance in firm-level, relative to industry-level, conditions. As a bonus, the blackboard game method of coordinating outputs also leads on average to firms with the lower costs having the larger market shares since the low cost firms will have the most to lose in rationing output, and thus announce higher planned levels of output. Of course, since only talk is involved, no formal contract is necessary. Thus we are not considering the case of formal cartels alone. Rather, the kind of commitment involving output intentions we are to consider would tit easily into an informal system of industry level planning. It is not our suggestion that this specific kind of information transmission is especially prevalent. However, part of our analysis will show the conditions under which it might flourish. In section 2 a simple no-fat model is described and the coordination blackboard game is specified and analysed. Proposition 1 describes a rational expectations solution, essentially finding the relationship between final output plans, the firms’ private information and other firms announcements. Section 3 describes the nature of equilbrium and numerical analysis is used to demonstrate the scope for profit advantages over the simple Cournot-Nash equilibrium with simultaneous output decisions. Conclusions are presented in a final section.
2. The single period quantity setting game Model specification
We will consider only an n-firm symmetric demand for a homogeneous product:
case with linear
market
p=u'-Q, where p is product price and Q is market supply. Demand and cost conditions are random and, to exclude inter-period learning, are assumed to be independent from one period to the next. The intercept a’ is a random variable, normally distributed N($,az). Variable costs of production for the ith firm are
ci(qi)
=
tm
+
‘lil4iv
where 4i is the ith firm’s output
and vi is a random
variable,
normally
M. Hviid and N.J. Ireland, Model consistent beliefs with information transmission in oligopoly
235
distributed N(O,a,2). The variance of vi is invariant across firms in order that firms are ex ante identical. The vi’s are assumed uncorrelated across firms and from one period to another so that above expected costs in one firm in one period does not imply above or below expected costs in any other firm or period.’ To simplify notation define ZE sc’-m. Then E(a) =di’-m=oZ, so that a-N(oS,a%). We shall assume that each firm receives perfect information (a perfect signal) concerning its own cost (rli) prior to choosing a strategy, but no direct information concerning other firms costs. Each fu-m also receives an imperfect signal Si on the realisation of a prior to choosing a strategy. The signal is assumed of the form Si =
CL+ &i,
where .si is normally distributed N(0, df), and E(si Ej) =OVi, ji # j. Further assume that a,qi and si are uncorrelated. In the absence of any other information, firm i would use (2) to predict the realisation of z, E(alSi). Given the normality assumption, ’ E(zlSi) is the convex combination of the signal Si and the prior (r:‘j E(aISi)=i+t(Si-E),
where
We will consider three distinct equilibrium concepts in the one period game. First the Cournot-Nash equilibrium reflects the situation of no information transmission, when firms simultaneously determine their outputs. Then we consider a collusion equilibrium where firms exchange their Si information (but not their lli), and then determine the best common output level. Obviously this requires some validation procedure which would not normally be available. Finally we use these two ‘polar’ cases, in terms of information sharing, to contrast a third equilibrium concept. This will be an equilibrium in what will be called a ‘blackboard’ game. ‘More generally, z can be interpreted as a global shock and r~ as a local shock. We have found it more natural to link the global shock to demand and the local shock to costs, but in our linear-quadratic case both could, by suitable change of sign, be linked solely to either demand or costs. sEq. (3) is true not only for normal errors but also for a range of other distributions; see Ericson (1989). 61n general, from Spanos (1986, p. 121-127). let X, Y be bivariate normal distributed with means and variances pi, UT. and covariance ul?. Then the correlation coefficient is p = uJ(u~u,), and the conditional expectation of X for Y=y is given by E(XI Y=y)=p,+p(y-y)u,/u,. Replacing y by Si using (2). and x by a yields (3) and (4).
236
M. Hciid
and N.J. Ireland,
Model
consistent
behefs wth
information
transmission
in o1igopol.v
Cournot-Nash equilibrium If the firm had no other information, firm i would choose its output qi to maximise expected profits, given ‘li and Si, but without knowledge of other firms information or output decisions:
where Q_i is the output of all the other firms in the industry. The first order condition of (5) is given by: qi=)[aT+t(Si-_)-_i-E(Q_i(S,)].
(6)
To solve (6) firms have to form model consistent expectations about how other firms react to their own private information (recall it is common knowledge for firm i that the other firms receive private information of a specific form given in (2) even though the realisation of Sj and ‘lj is unknown). Let this expectation be given by: qj=p~++r(Sj-~)+6s’
(7)
I’
Implying that, as E(S,-itSi)=
t(Si-oS) and Eq,=O,
E(Q_ilSi)=(n-
l)(Si-e).
I)p~+rt(n-
(8)
Inserting (8) in (6) we obtain:
As the expectation has to be model consistent we require that (7) is of the same form as (9) for all i, j, allowing /3,y and 6 to be identified, and implying that
and thus industry output is given by
Q=nE/(n+ l)+tZ(Si-E)/[2+(n-
l)t]-Z&
where all summations are from 1 to n. Firm i’s ex ante expected profit is given by E((a-vi-Q) qij where qi, Q are given above. Substituting and taking expectations yields (after using (4) to substitute out the of term): E(ni) =
Z2/(1 + n)2 + tO,2/[2 + (n - l)t12 + *ai.
(10)
M. Huiid and NJ.
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beliefs with information
transmission
in oligopoly
237
Collusion
With private information, what can be achieved through collusion depends on what (if any) part of the information can be shared or verified. Note that after receiving the private information on costs firms are no longer identical. To avoid problems concerned with the revelation of information when all output is allocated to the lowest cost firm [see for example Cramton and Palfrey (1990)] we assume that only the individual firm’s signals Si are shared. In this case the industry decides the best equal output per firm. Thus output per firm, q, is chosen to maximise
where the composite signal S* =a+ Z&i/n summarises the information available to all firms. Expected equilibrium profits are given by: E(ni)=(E’
+ t*e,2)/4fI,
on a
(11)
where c* = az/(oi + az/n). In this benchmark case, expected profits per firm are unaffected by variations in costs (because of the constant shares assumption). For n> 1 the coefficient of 0.’ can be shown to be larger than the coefftcient in the Cournot-Nash case. Thus the collusion allows for better use of the demand variability due to the pooling of information. Blackboard
game solution
To contrast with the extreme equilibrium concepts applied above, we will consider an equilibrium where information is imperfectly pooled via credible output plans. As suggested in the Introduction, an intuitive grasp can be obtained by thinking of a blackboard where output plans are posted. Plan revisions take place until no firm wishes to revise its plan given that other firms’ plans will be enacted. An extensive form representation of the plan revision game is shown in fig. 1. Starting at A with stated plans qi, Q_i, where the choice of i is arbitrary, firm i has to revise its plan without knowing whether other firms will also revise their plans. Thus firm i does not know which node in the information set B is appropriate. Given the information in Q-i3 firm i decides whether to change its plan to qi or leave it as it is. If other firms do not change their plans, then a decision not to change the ith firm’s plan means that each firm is happy to enact its plan given other firms enact theirs. If this is not the case at least one firm will change its plan and the game returns to A with updated plans. It is clear that each tit-m has to act as if it is the firm’s last chance to revise its plan. Thus as the only equilibrium is one where no firm is unhappy with its plan, there is no payoff from procrastination, and each firm will simply set its plan
238
M. Hviid
and N.J. Ireland,
Model
consutent
beliefs with information
transmission
in oligopol)
produce conlrary to plan. Suspend blackboard game
Produce Pi* Q.i proceed to next repetition
Q-i
go back to A with updated information based on new plans
Fig. 1. Output plans in the blackboard game.
at that output, given all its information embodied in Q_i, which will maximise its expected profit. However the information provided by the ith firm’s output plan leads to an equilibrium which departs from the standard Cournot-Nash equilibrium. It is the extra expected profit from this equilibrium compared to the Cournot-Nash equilibrium that allows the firms to be disciplined in terms of keeping to their last announced plans. Deviations are punished by the industry suspending the blackboard game and reverting to Cournot-Nash for a sufficient number of periods to make deviating unprofitable almost certainly. An equivalent punishment to suspending the blackboard game is simply the natural one of the removal of output plan credibility. The information transmission and payoffs from the blackboard game will be investigated below within the framework of a rational expectations equilibrium using the technique known as undetermined coefficients. In order to decide on an output level, a firm must form an expectation concerning how the rest of the industry’s beliefs will be affected and how this will affect other firms’ behaviour. This will depend on the firms’ own information as well as the information and market pre-emption message incorporated in the ith firm’s planned output. Let the conjectural function held by the ith firm be
M.
Hviid
and NJ.
Ireland,
Model
consistent
beliefs
with in/ormnrion
Q_i=a+bS’_i+cqi.
transmission
in oligopoly
239
(12)
where S’_i is the ‘aggregate’ signal (to be defined below) on which the other firms base their choice of strategies. Note that (12) assumes that plans have already been chalked up on the blackboard; initially plans are made given only each firm’s private information. Thus the conjecture is solely on the effect of observable plans. The form of (12) reflects the fact that if firm i observes Q _ i it would be able to infer additional information on a by solving for S’-i. The constancy of the conjecture c is obviously a limitation.’ However, it should be noted that the blackboard game allows full investigation of responses so that local beliefs must accord to global conjectures. Also the symmetry of the model suggests that a natural form of the conjecture should reflect the form of Cournot reaction functions. Due to this structure of the model, the first-order condition for firm j must be linear in Sj and vi. Hence Q_i must be linear in all S, vi, jS i. The signal on a obtained from Q-i can thus be written as: S’_i =
C (Sj-kqj)/(n-
1)~
C SJI(nj#i
j+i
=x+ 1 (Ej_kqj)/(n_l)=a+8i,
1)
(13) (13’)
j#i
where
SJ=Sj-k~j~
Bi=
C
(Ej-kqj)/(n-
1)
j#i
and k is a parameter which reflects the relative importance of Sj and 9j on other firms’ outputs. Observing Q-i only partially reveals the information held on a by the other (n- 1) firms (the Sj) due to the ‘noise’ arising from random costs (qj); only the S; are inferred. As firms have to announce their output plans prior to carrying them out, the quantity Q-i is available to the ith firm. From its conjectural function (12), S’-i can then be inferred. Thus any candidate for an equilibrium set of output plans ql, q 2,. . . ,qn must incorporate the information contained in S’_1, s- 2,. . . , S’_, respectively. Each firm i has two signals, Si and S’_i to use for predicting the demand parameter a. Firm i’s information on a given in (2) and (13’) can be summarised, see Jaffe and Winkler (1976, p. 50, 57-58), as
‘See Boyer and Moreaux (1983a. b).
240
M. Hviid and N.J. Ireland, Model consistent beliefs with information transmission in oligopoly
Si=6Si +( 1_6)S’_i,
(14)
VAR(Si)=of
(19
+sof,
where 6 = VAR(ei)/[ VAR(Ei) + VAR(ei)] =(e,Z + k2ei)/(qef + k2ai)
(lo)
E(~i1Si,rli)=[(tSi+(l_t)di-_i_Q)qi]
(17)
and
subject to (12). Note that (5) is now the special case in which 6= 1, i.e., the case where either crf 3 0, in which case Si contains sufficient information on a or CJ:+ cc in which case S’_i is an infinitely noisy signal and thus has no value. 3. Equilibrium
with model consistent expectations
We define an equilibrium with n active firms as follows: Definition 1.
An n-tuple(q,,q,, . . . , q,,) is an equilibrium if, given information (Si,qi) no firm i wishes to change qi, where S’-i=(Q-i-a-cq)/b and (a, b,c) form a consistent conjecture.
In such an equilibrium no firm wishes to change its plan. Of course each firm tries to manipulate the oligopoly process by choosing the strategically most advantageous output plan: however, other firms can interpret such rational behaviour and infer the correct market signals. The complete equilibrium process of the game depicted in fig. 1 incorporating information transmission can thus be summarised in the following. (1) Each tit-m states it best output plan given its conjectural function (12) of competitors’ reactions, and given its private information. (2) Any firm i can revise its output plan given the information S’_i which is extracted from the initial plans.
M. Hviid and NJ.
Ireland,
Model consistent
beliefs with information
transmission
in oligopoly
Z-t1
(3) No further revisions are desired and the plans are put into effect. If any firm ‘pretends’ information at stage 1 or 2 then further iterations are required to attain the same equilibrium. Alternatively the pretending firm has to adopt a sub-optimal plan and realise less expected profit. Thus the above sequence is sub-game perfect. Of course, the simplicity of the outcome is due to the binding nature of final plans. A last-minute change of plan by one firm means that all other firms will still be able to adjust their behaviour: talk is not cheap, This can be compared with Mailath (1989), where the ‘talk’ is replaced by real outputs.* We show in the Appendix that there exist two sets of coefficients a, b,c, such that the conjectures embodied in (12) are consistent with behaviour dictated by maximising (17). When firms all hold common consistent conjectural functions, responses to variations in firm behaviour correspond to that predicted by the conjectural function. Proposition 1 below focuses on the coefficient c( =dQ_,/dq,) in the conjectural function (12), since this is paramount in determining the observed degree of competition in equilibrium. Proposition 1. There are two consistent conjectural functions. In one case the coeficient c= - 1, and this is the competitive conjecture. In the other case c=c* where c*=(2-(n+l)J*)/b* and s*~[l/n,l] is uniquely determined by (13), (4’) and k=(t6)-‘. Proof.
See appendix.
Since the competitive conjecture is inferior in terms of pay-offs, we will concentrate our attention on the c* conjecture.g A number of important properties of the c* conjecture can be stated immediately from Proposition 1. (1) c* is an increasing function of 0,2, a decreasing function of 0, and an ambiguous function of r_rtand n. (2.i) As ~~+O,c*+n-1. (2.ii) As C: + af/[f(n + 1) + o:/o,~], c* + 0. (Ziii) As C: +@,c*+ - 1, where @=[n/(n-2)]az/[f(n+ l)+of/az]*. (3) IfO
242
M. Hciid
and N.J. Ireland,
Model
consistent
belie>
with information
transmission
in oligopoly
The sign of the c* conjecture in Proposition 1 relates to output strategies incorporating information transmission. Thus more uncertainty over the prior predictor of a (higher ai) will lead to a higher value of c* as firms put more reliance on market signals as a whole. Similarly as c,’ increases, indicating less information content in output plans, so firms rely more on their own private signals and the coherence of the market is reduced. The effect of increased variance in the private signal (higher 0:) is two-fold; first it leads to more reliance on market signals relative to private signals, but secondly it leads to more reliance on the prior predictor so that other firms’ outputs become less important for their information content concerning market demand, and relatively more important as indicators of market supply. Similarly, an unambiguous effect from increasing n is not forthcoming. The specific solution 2.i above deserves further comments. In this case it is as if firms simultaneously collude over their choice of quantities (c* = n - 1) and pool their information (6* = l/n). With 0,: close to zero, Q-r becomes a sufficient statistic for Cj~i Sj and the apparent pooling of information follows using Theorem 1 in McKelvey and Page (1986). In this special case market information is imparted fully and involuntarily through announced plans, and the collusion equilibrium is obtained. For some parameter values, expected profits may be small and even negative. In these situations we would expect either some firms to exit the market or that the blackboard game would break down and be replaced by a simple Cournot-Nash supply process, or both. In our discussion below, we ignore these and other complications by assuming that the number of firms are fixed, that the blackboard game is viable and that all firms always produce positive quantities in equilibrium. Some examples of parameter values supporting the discussion are given at the end of this section. Output in equilibrium
Solving for the three parameters in (12) for the c* conjecture case we can write down the reaction function of firm i as: qi = {(&I-
1)&?+(6n-
+[2-(n+l)E]Q_i}/G
l)st(s:-E) i=l,...,
n,
where G=(n- l)a+(n-2)[2-(n+ 1)6]. Tedious manipulations following proposition concerning outputs and expected profits.
(18)
give us the
Proposition 2. In an equilibrium with consistent conjectural variation c*, aggregate output Q, individual output qi and expected profits E(ni) of $rm i before the signals are received, are respectively
M. Huiid and N.J. Ireland,
Model
consistent
beliefs with
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transmission
in oligopoly
243
(i) Q=+IcG++&Z(SJ-E)
(19)
(ii) 4i=tscr+A(SI-a)+(f6t-A)~(SJ-a)/n,
(20)
where A=6t(6t-l)/[(n-1)(2-6n)]>Ofor
l/n<6<2/n,
(21)
(iii) E(~i)=~{6(2-n~)o72+t~(2-n~t)a,Z-(f6t)za~+[26/(2-n~)-1]a,Z},
(22)
providing all n firms produce at positive output levels.” Proof.
See appendix.
The equilibrium associated with the c* conjecture described in 2 is a function of all the information in the supplying industry. assumed qi >O Vi,E(ITJ >OVi is also implied in the absence of Further intuitive feel for the equilibrium can be obtained by expected values of the outputs and price-cost margins.
Proposition As we have fixed costs. considering
Proposition 3. Ex ante of any signals being receiced, the c* conjecture yields an equilibrium with the following characteristics (i) The expected industry output is E(Q)=fn& and is increasing in n, ti, and and decreasing in of. (ii) The expected output per firm is E(q) = $%i and is increasing in 5 and o: and decreasing in n and of. (iii) The expected price-cost margin is E(p- m) = ji( 1 - +n8) and is increasing in d and oi and decreasing in n and o:.
a,f,
Proof. E(p-m)
E(Q) and E(q) are found by taking expectations
is found by taking expectations of eq. (A.ll) Differentiation yields the comparative static results. 0
of (19) and (20). in the Appendix.
Proposition 3 shows that the average effect of greater c,’ or smaller 0: is to move the industry towards a more coherent structure with associated lower industry supply and higher price-cost margins. Numerical results and enforcement
A full solution to the equilibrium can be found only by numerical methods “To see why the assumption that all n firms produce at a positive level is necessary, note that and qi+co if Si-a>p, a/n+tp/n if S;-z=p, or --co if S;-zcp, where p=Zj(S;-a)/n. Further, from Proposition 1(2.iii)6-+2/nac* + - 1. Thus as we approach the competitive conjecture, the output levels given in (20) are not bounded. As 6+2/n, only the firm with the most encouraging signals will produce and supply the whole market. 6-r2/n=A+cO
244
M. Huiid and NJ.
Ireland, Model consistent
Table The numerical ”
z or z ?n : C*
E(n) E(CN --II) E(Col-fI)
results:
beliefs with information transmission
in oligopoly
1
5= 10 and 0:=4,
2 4 0.01 0.5055 2.978
2 4 0.10 0.5456 2.833
2 4 0.40 0.626 2.597
0.9566 12.83 12.00 12.83
0.6658 12.73 12.03 12.85
0.1944 12.10 12.10 13.08
E(CN-U) is protit in the Coumot-Nash collusion case.
( =8 in col. 5).
0.10 0.5813 1.970
2 4 0.10 0.5330 2.376
0.4407 12.61 11.54 12.84
0.7526 13.25 11.92 13.31
2
1 I
case: E(Col -l7)
5 4 0.10 0.2769 4.612 1.223 4.410 2.927 5.168 that in the
since 6* is only defined implicitly. It is important however to examine how the expected profit from playing the blackboard game exceeds that of simple Cournot-Nash strategies [see (IO)]. Without such a premium there could be no discipline in enforcing firms to produce according to their final announcements, and the blackboard game would have no credibility. The numerical analysis is summarised in table 1. Numerical results comparing the expected profits of the coordination game and the Nash game. The first three columns consider a duopoly with r~f=a: =4. The amount of noise in information transmission is measured by G: and this increases across the three columns. When it is insignificant (0.01) expected profits from the blackboard game approximates the ‘collusion’ expected profits [eq. (1 l)]. When it is 0.40 there is significant noise for the advantage of the blackboard game over a Cournot-Nash game to disappear. Notice that this advantage disappears at a positive conjecture (c*z 0.19). This reflects the fact that the Cournot-Nash equilibrium yields higher expected profits as cost and demand variations increase: the blackboard game lowers this advantage and only performs better than the Cournot-Nash when cr,’ is smaller than the value which would yield c* =O. When 0: =O.lO, over 80% of the loss from Cournot-Nash compared to the full collusion is made up by playing the blackboard game. These examples show how important the noise r~,’ is in determining the degree of monopoly in the equilibrium. If G,’ is large then firms do not extract much information concerning market demand from the blackboard game and so individual firms only have an incentive to pre-empt the market, leading to high outputs and low prices. Columns 4-6 in table 1 consider the effect on the equilibrium of a better signal (smaller crz), more random demand (higher g.‘) or more firms. All cases can be related to the blackboard case in column 2. A better signal reduces expected profits, and more randomness increases profits, due to the convexity of profits in a. The pattern of the blackboard game yielding most of the difference in expected profits from the collusion case relative to the
M. Hviid and NJ. Ireland. Model consistent beliefs with information
transmission
in oligopoly
245
Cournot-Nash case is repeated. When there are more firms, expected profit is of course lower but the conclusion remains. 4. Conclusions An explanation for the cohesiveness of market supply has been advanced which is based on the transmission of information via quantity plans. Firms take account of the way information will be extracted from plans and used by the rest of the industry. Central to the process is the key conjecture c* which is determined by the underlying parameters of the model (see Proposition 1). It increases as a firm’s prior estimate becomes less reliable (af increases) and as information transmission becomes less noisy (r$ decreases). Thus as the information content of quantity signals become more needed and more efficient, so the industry becomes more collusive in character, and (from Proposition 3) expected industry output decreases and expected pricecost margins increase. Although other contexts have been suggested, an interesting view of our model of a blackboard game is found within a supergame. Because, in each repetition, firms receive some private information about the current realization of the random variables, it is not desirable to specify output for future periods. Imagine that at the beginning of each period firms can meet and talk. Private information cannot be transmitted by direct communication of market demand predictions because truthful revelation of such private information cannot be verified. However, if outputs are observable,” output plans could be made a credible means of information transmission, by punishing deviations from the set of plans to which firms are committed. This transmission would typically not be perfect, but the better the extraction of information (the smaller the ‘noise’ of lirm-specific variations in costs), the more cohesive the industry’s apparent organization and the nearer is price to the monopoly price. Only when there is no cost uncertainty can the monopoly outcome be implemented. In short, the bounds on implicit monopolisation of the industry are set by both the means and the efficiency of market information transmission. Many assumptions, such as linear demand, homogeneous products and constant costs, contributed to the tractibility of the analysis. The central limitation however has been that parameter values were such as to provide a profit premium from playing the blackboard game, and hence provide the mechanism for enforcing the rule of the game: that only current announced plans can be put into effect. The numerical results in table 1 showed examples of acceptable parameter values. Of course, even if the gains from “The case when outputs Abreu, Pearce and Stachetti
are not observable (1985).
is considered
by Green
and
Porter
(1984) and
246
M. Hviid and N.J. Ireland, Mode/ consistem beliefs with information rransmission in oligopol)
retaining for future periods the coordination implicit in the blackboard game are large, breakdown might still be observed. This would be caused by extreme high expected profits in the current period outweighing the gain from conforming to the rule. Such a possibility cannot be excluded since r, ~~ do not have a finite upper bound on their supports, although it would become more remote as the discount factor on the future declined. That the coordination and information transmission may break down in extreme circumstances probably reflect a general feature of games of this kind. The model outlined in this paper could be extended to relax a number of assumptions in order to apply the underlying thesis to a wide selection of problems. Possible examples might include an extension to monopolistic competition where conjectural variations are a determinant of whether too many or too few products are produced in a Chamberlinian equilibrium [see Koenker and Perry (1981) and Ireland (1983)]. Further extensions relate to the inclusion of risk averseness of firms, where information transmission would have the secondary effect of reducing risk. It is hoped to explore these extensions in future research.
Appendix Proof of Proposition I. Writing Q = Q _i+ qi, the first order condition for maximising (17) with respect to qi given the conjectural function (12) the combined signal (14) and pi, is
Substituting S’_i from (12) into (A.l) and solving for qi in terms of Si, rli, Q-i and Cryields the reaction function, incorporating reaction to information as q,=(l-t)!X&zar(l-@+ 1 2+c+cr(l-6) where S:=S,-(
tab
2+c+ct(l-6)
S!+ t(l--4-b ’ 2+c+ct(1-6)
Q
ir
(‘4.2)
l/tS)rli implying, using (13’) together with (4’) that
(A.3)
Note that a unique value of k greater than one exists to solve (A.3). Using (16) and (4’), the right-hand side can be written as
M. Huiid and NJ.
not +
Ireland,
Model
consistent
belitfs with information transmission
in oligopoly
247
kza2 a2 “+A.
af
a,2+k2a,2
This is a decreasing function of k, and for all k, strictly greater than one. Further it is bounded at 1 +~:/a: as k -+ m. Therefore (A.3) has a unique positive solution and we will simply denote this as k. A reaction function of the form (A.2) holds for any ith firm. We must now investigate the joint response of all other firms, given their reaction functions (A.2), to an increase in one firm’s output. Following Perry (1982) we not only take into account the direct effects but also the interaction among all the other firms (if firm i and i react to a change in the output of firm 1, then they also react to each other’s reaction). Without loss of generality we focus on firm 1. For i# 1, we can write (A.2) as t(l-6)-b %- 2+c+ct(l-6) t6b
+2+c+ct(l-6)
Q__ _(l-t)bE--at(l-6) ‘.t2+c+ct(l-6) p+ t(l--4-b I 2+c+ct(l-6)
41y
(A.4)
where Q-i.1 is the sum of outputs of all but the ith and the first firms. Summing (A.4) over i from 2 to n yields
t6b +2+C+Ct(l-6)
i S;+(n_l) i=2
r(l-4-b 2+c+ct(l-6)q1*
(A.9
The actual response function (AS) has the same form as the conjectured response function (12). If the values of a, b and c in (12) can be found such that (12) and (A.5) are identical, a consistent conjectural variations equilibrium has been identified. Equating the coefficients in (12) and (A.5) we get [(l-t)bl-r(l-6)a](n-1) ‘=b(2+c)+ct(l-6)-(n-2)(r(1-6)-b)’ b=
(n- l)t6b b(2+c)+ct(l-6)-(n-2)(t(l-6)-b)’
(A-6)
(A.7)
248
M. Haiid and N.J. Irelund, Model consistent beliefi with information
transmission
in oligopoly
(n-l)(t(l-@--b) C=b(2+c)+ct(l-6)-(n-2)(t(l-S)-b)~
(‘4-Q
Take the ratio of (A.7) to (A.8) and solve to get b=t(l-6)-t&.
(A.9)
Insert this in (A.8) and rearrange to get sc2--[2-(n+2)6]c-[2-(n+1)6]=0.
(A. 10)
The two roots of (A.lO) are the conjectures given in the proposition. From (A.9) and (A.@ we see that for each of the two values of c, there is a unique value of a and b. 0 Proof of Proposition 2. Part (i) and (ii) are found by solving (18). To show (iii), note that the price-cost margin for the ith firm is cc--Q-vi. Using Q from (i) above a, q and qi
allows it to be written
p-m--?fi=a-+l&!-+&
i
purely
in terms of the random
variables
(s;-di)-~i.
(A.1 1)
i=l
Combine (ii) and (A.1 1) and take unconditional
+6cr+A(S;--q+
x [
where A is defined (iii).
in
the
expectations
to get
1 1j$l II )&-A
proposition.
;
Solving
(SJ-6)
and
,
rearranging
yields
0
References Abreu, D., Journal Boyer, M. Journal Boyer, M. theory:
D. Pearce and E. Stachetti, 1985, Optimal cartel equilibria with imperfect monitoring, of Economic Theory 39, 25 l-269. and M. Moreaux, 1983a. Conjectures, rationality and duopoly theory, International of Industrial Organization 1, 2341. and M. Moreaux, 1983b. Consistent versus non-consistent conjectures in duopoly some examples, Journal of Industrial Economics 32, 97-l 10.
M. Hviid
and ‘V.J. Ireland,
Model
consistent
beliefs with information
transmission
in oligopoly
249
Bresnahan, T.F.. 1981, Duopoly models with consistent conjectures, American Economic Review 71. 934-945. Cramton, P.C. and T.R. Palfrey, 1990, Cartel enforcement with uncertainty about costs, International Economic Review 31. 17-47. Ericson, W.A., 1969, A note on the posterior mean of a population mean, Journal of the Royal Statistical Society 31, 332-334. Frisch, R., 1933, Monopole-polypole, la notion de force dans I’Cconomie, Nationalakonomisk Tidsskrift 71, 241-259. Green. E. and R. Porter, 1984, Non-cooperative collusion under imperfect price information, Econometrica 52, 87-100. Ireland, N.J.. 1983, Monouolistic competition and a firm’s product range, International Journal of Industrial Organizaiion 1, 239-?52. Jaffe, J.F. and R.L. Winkler, 1976, Optimal speculation against an efficient market, Journal of Finance 31, 49-61. Kalai, E. and W. Stanford, 1985, Conjectural variations strategies in accelerated Cournot games, International Journal of Industrial Organization 3, 133-152. Koenker, R.W. and M.K. Perry, 1981, Product difTerentiation, monopolistic competition and public policy, Bell Journal of Economics 12, 217-231. Mailath, G.J., 1989, Simultaneous signalling in an oligopoly model, Quarterly Journal of Economics 104,417-427. McKelvey, R.D. and T. Page, 1986, Common knowledge, consensus and aggregate information, Econometrica 54, 109-127. Perry, M.K., 1982. Oligopoly and consistent conjectural equilibria, Bell Journal of Economics 13, 197-205. Rotemberg, J. and G. Saloner, 1986, A supergame theoretic model of business cycles and price wars during booms, American Economic Review 76, 39m7. Ulph, D., 1983, Rational conjectures in the theory of oligopoly, International Journal of Industrial Organization 1, 131-154.
Model consistent beliefs with information transmission in oligopoly* Morten
Hviid and Norman
Department
of Economics, University
J. Ireland
of Warwick,
Coventry,
CV4
7AL,
UK
Accepted for publication November 1991
A model of market coordination in the presence of private information is developed. One component is a period of talk about output levels. The talk ends when no firm wishes to change its latest announced planned output level, and only if all firms enact their latest announced plans will such talk be repeated in future periods. It is shown how the degree of coordination and apparent collusion achieved is related to the noise in signalling market information.
1. Introduction
The ability of firms in an oligopolistic industry to reach agreements to restrict production levels and enhance profits is based on two key factors. First, it must be possible to observe (at least to some extent) whether or not individual firms cheat on their agreements. Second, a disciplinary device must be available to punish cheating, so that the agreements are incentive compatible. From the folk theorems for repeated games we know that such discipline is available through an infinite repetition of the stage game, so that short-term profits from cheating are outweighed by punishment in later periods. The problem of non-observability of firms’ actions limits the extent of collusive agreements.’ This complication is especially limiting if uncertainty is (partly) resolved before firms choose their individual strategy. An example of this is discussed in Rotemberg and Saloner (1986).’ A related literature [founded by Frisch (1933)] focusses on the concept of lHviid gratefully acknowledges financial support from the Danish Social Science Research Council. Preliminary versions of this paper have been presented at seminars in the Universities of Aarhus, Copenhagen, Sheffield and Warwick, and at Birkbeck College, London. We are grateful to participants of these seminars for many helpful suggestions and comments. Particular thanks are due to Torben Andersen, Paul Geroski, Xavier Vives, and anonymous referees of this journal. ‘See for instance Green and Porter (1984). ‘In their model, lirms know the current realisation of the random variable before they choose their current strategy, i.e., whether or not to comply with the collusive arrangement. The collusive arrangement, on the other hand, is made before the realisation and is based on expected values. Similarly the consequence of a punishment phase is based on expected value. Thus there exist sufficiently good realisations for deviations to be profitable. 0176-2680/92/505.00 0 1992-Elsevier
Science Publishers B.V. All rights reserved
232
,M. Hciid and N.J.
Ireland,
Model
consistent
beliefs wirh information
transmission
in oligopoly
output response conjectures. This literature has attempted to determine the degree of apparent collusion in an industry by asking what conjectured output reponses are consistent with actual responses [see for instance Boyer and Moreaux (1983a, b), Bresnahan (1981), Perry (1982) and Ulph (1983)]. The outcome has been disappointing. Either too limited a set of conjectural equilibria have been found to account for observed variations, or it has been claimed that no refinement can be achieved, and that any equilibrium is a consistent conjectures equilibrium for some local conjectural function [Boyer and Moreaux (1983a)]. This literature has not encompassed questions of information and observability; nor has it in general incorporated dynamic aspects of repeated interaction. 3 It seems necessary therefore to shift ‘the emphasis . . . from the equilibrium concept itself to the origins, development and validation of conjectures’ [Bayer and Moreaux (1983, p. 40)]. In this paper we go some way towards offering an endogenous explanation of conjectures in terms of the ability of firms to signal market information. Further, the level of current period coordination is supported by expected profits in future periods. We focus on coordination within a market which is affected by stochastic shocks in each period. Then market shares or output quotas cannot be allocated without common agreement concerning the size of the market. We would naturally expect each firm to have its own view as to this size, perhaps formed from its own private information obtained from its market contacts. We will argue that the pooling of such private information is fraught with ditliculties. In particular, if firms are heterogeneous with respect to their costs of production (and particularly if a firm’s cost level is also private information) then firms will have an incentive to provide an information input to the industry cartel which is biased by their own cost positions. If a firm anticipates higher costs than faced by other firms (perhaps because of employees’ bargaining power) then it might claim that the market will be very small, thus trying to reduce all firms’ output quotas and change the balance between high price and sales volume to favour its own cost position. Relying on individual firms to supply information which has no public observable source and is thus unveritiable is not satisfactory. The problem is that talk is ‘cheap’; the firm does not have to behave in any way consistent with the information it supplies. It does not necessarily reveal either its true costs or its true initial market information. Furthermore the level of punishments is more difficult to judge as the extent of deviations from truth telling cannot be observed. As an alternative to such cheap talk, consider the following. Firms may submit output plans which can be modified after listening to other firms output plans. Imagine a blackboard where each firm’s initial plan (a size of 3An exception
to the latter is Kalai
and Stanford
(1985).
M. Hoiid and N.J. Ireland,
Model
consisrenr
beliefs with information
fransmission
in oligopoly
233
output) is chalked up. Then any firm can erase and revise its plan. When no firm wishes to revise its plan further, given all other firms’ plans, these are the outputs adopted by the industry. Actual outputs are observable and any deviation from Sinai planned outputs are punished in later periods. Basically each firm is forced to put its money where its mouth is. In a full information version of the above game many outcomes may be supportable as perfect equilibria depending on an appropriate disciplining device in the overafl supergame. In a model with imperfect info~ation, the concept of the ‘right’ industry output or the ‘right’ output level for a particular firm is not well defined, since information transmission wilt always be manipulated by firms for their own private gains. However, it is still possible to discipline firms to reveal their final output intentions, and then to implement them, provided that the game played ‘with the blackboard’ yields more expected payoff than that without such coordinating interaction. Basically, true output intention is a verifiable and significant part of the firm’s behaviour. It contains information. Not only is it a way of committing to supply the market (in the Cournot-Nash sense) but it is also a way of indicating a view of the potential market size. No firm wishes to give the impression it believes that demand is high since this would encourage high output decisions from other firms. On the other hand low output plans imply that more optimistic firms would average the pessimistic signal with their own information and tend to achieve a higher market share. Thus when a holiday tour operator states that it plans to sell, say, f million holidays in the next year, this is partly market preemption and partly an indication of that firm’s expectation of the market size next year. The latter information can be at least partially inferred since output plans which are to be implemented are not ‘cheap talk’. They involve actions which involve profits or losses. That ‘tinal’ plans have to be implemented means that, although the planned market supplies can be changed in response to other firms supply plans, they cannot be changed secretly, nor can they be changed after other firms plans are committed. In the holiday tour industry, secrecy is unlikely. More importantly the number of holidays provided becomes public information. Thus cheating will be found out and will lead to coordination breaking down in future periods. This is a disincentive to cheat provided that the blackboard game yields on average a reduction in total supply and more monopoly pro&s. The information signalhng aspect of the blackboard game depends on how far the market conditions observed by one firm are relevant for the profitability of others. If each tour operator sold a rather different type of holiday to other operators, there would be less to be learned. We model this ‘noise’ as firms differing with respect to an independent random variable on their costs, but this is just for clarity. The key point is that, as the noise becomes more significant, the information motive beomes less important and
234
hf. Hviid and N.J. Ireland, Model consisrent beliefs with information
transmission
in oligopoly
each firm simply tries to preempt as much of the market as possible, leading to a more competitive outcome. In this way the degree of monopoly observed in an industry may be determined by the relative importance in firm-level, relative to industry-level, conditions. As a bonus, the blackboard game method of coordinating outputs also leads on average to firms with the lower costs having the larger market shares since the low cost firms will have the most to lose in rationing output, and thus announce higher planned levels of output. Of course, since only talk is involved, no formal contract is necessary. Thus we are not considering the case of formal cartels alone. Rather, the kind of commitment involving output intentions we are to consider would tit easily into an informal system of industry level planning. It is not our suggestion that this specific kind of information transmission is especially prevalent. However, part of our analysis will show the conditions under which it might flourish. In section 2 a simple no-fat model is described and the coordination blackboard game is specified and analysed. Proposition 1 describes a rational expectations solution, essentially finding the relationship between final output plans, the firms’ private information and other firms announcements. Section 3 describes the nature of equilbrium and numerical analysis is used to demonstrate the scope for profit advantages over the simple Cournot-Nash equilibrium with simultaneous output decisions. Conclusions are presented in a final section.
2. The single period quantity setting game Model specification
We will consider only an n-firm symmetric demand for a homogeneous product:
case with linear
market
p=u'-Q, where p is product price and Q is market supply. Demand and cost conditions are random and, to exclude inter-period learning, are assumed to be independent from one period to the next. The intercept a’ is a random variable, normally distributed N($,az). Variable costs of production for the ith firm are
ci(qi)
=
tm
+
‘lil4iv
where 4i is the ith firm’s output
and vi is a random
variable,
normally
M. Hviid and N.J. Ireland, Model consistent beliefs with information transmission in oligopoly
235
distributed N(O,a,2). The variance of vi is invariant across firms in order that firms are ex ante identical. The vi’s are assumed uncorrelated across firms and from one period to another so that above expected costs in one firm in one period does not imply above or below expected costs in any other firm or period.’ To simplify notation define ZE sc’-m. Then E(a) =di’-m=oZ, so that a-N(oS,a%). We shall assume that each firm receives perfect information (a perfect signal) concerning its own cost (rli) prior to choosing a strategy, but no direct information concerning other firms costs. Each fu-m also receives an imperfect signal Si on the realisation of a prior to choosing a strategy. The signal is assumed of the form Si =
CL+ &i,
where .si is normally distributed N(0, df), and E(si Ej) =OVi, ji # j. Further assume that a,qi and si are uncorrelated. In the absence of any other information, firm i would use (2) to predict the realisation of z, E(alSi). Given the normality assumption, ’ E(zlSi) is the convex combination of the signal Si and the prior (r:‘j E(aISi)=i+t(Si-E),
where
We will consider three distinct equilibrium concepts in the one period game. First the Cournot-Nash equilibrium reflects the situation of no information transmission, when firms simultaneously determine their outputs. Then we consider a collusion equilibrium where firms exchange their Si information (but not their lli), and then determine the best common output level. Obviously this requires some validation procedure which would not normally be available. Finally we use these two ‘polar’ cases, in terms of information sharing, to contrast a third equilibrium concept. This will be an equilibrium in what will be called a ‘blackboard’ game. ‘More generally, z can be interpreted as a global shock and r~ as a local shock. We have found it more natural to link the global shock to demand and the local shock to costs, but in our linear-quadratic case both could, by suitable change of sign, be linked solely to either demand or costs. sEq. (3) is true not only for normal errors but also for a range of other distributions; see Ericson (1989). 61n general, from Spanos (1986, p. 121-127). let X, Y be bivariate normal distributed with means and variances pi, UT. and covariance ul?. Then the correlation coefficient is p = uJ(u~u,), and the conditional expectation of X for Y=y is given by E(XI Y=y)=p,+p(y-y)u,/u,. Replacing y by Si using (2). and x by a yields (3) and (4).
236
M. Hciid
and N.J. Ireland,
Model
consistent
behefs wth
information
transmission
in o1igopol.v
Cournot-Nash equilibrium If the firm had no other information, firm i would choose its output qi to maximise expected profits, given ‘li and Si, but without knowledge of other firms information or output decisions:
where Q_i is the output of all the other firms in the industry. The first order condition of (5) is given by: qi=)[aT+t(Si-_)-_i-E(Q_i(S,)].
(6)
To solve (6) firms have to form model consistent expectations about how other firms react to their own private information (recall it is common knowledge for firm i that the other firms receive private information of a specific form given in (2) even though the realisation of Sj and ‘lj is unknown). Let this expectation be given by: qj=p~++r(Sj-~)+6s’
(7)
I’
Implying that, as E(S,-itSi)=
t(Si-oS) and Eq,=O,
E(Q_ilSi)=(n-
l)(Si-e).
I)p~+rt(n-
(8)
Inserting (8) in (6) we obtain:
As the expectation has to be model consistent we require that (7) is of the same form as (9) for all i, j, allowing /3,y and 6 to be identified, and implying that
and thus industry output is given by
Q=nE/(n+ l)+tZ(Si-E)/[2+(n-
l)t]-Z&
where all summations are from 1 to n. Firm i’s ex ante expected profit is given by E((a-vi-Q) qij where qi, Q are given above. Substituting and taking expectations yields (after using (4) to substitute out the of term): E(ni) =
Z2/(1 + n)2 + tO,2/[2 + (n - l)t12 + *ai.
(10)
M. Huiid and NJ.
Ireland,
Model
consistent
beliefs with information
transmission
in oligopoly
237
Collusion
With private information, what can be achieved through collusion depends on what (if any) part of the information can be shared or verified. Note that after receiving the private information on costs firms are no longer identical. To avoid problems concerned with the revelation of information when all output is allocated to the lowest cost firm [see for example Cramton and Palfrey (1990)] we assume that only the individual firm’s signals Si are shared. In this case the industry decides the best equal output per firm. Thus output per firm, q, is chosen to maximise
where the composite signal S* =a+ Z&i/n summarises the information available to all firms. Expected equilibrium profits are given by: E(ni)=(E’
+ t*e,2)/4fI,
on a
(11)
where c* = az/(oi + az/n). In this benchmark case, expected profits per firm are unaffected by variations in costs (because of the constant shares assumption). For n> 1 the coefficient of 0.’ can be shown to be larger than the coefftcient in the Cournot-Nash case. Thus the collusion allows for better use of the demand variability due to the pooling of information. Blackboard
game solution
To contrast with the extreme equilibrium concepts applied above, we will consider an equilibrium where information is imperfectly pooled via credible output plans. As suggested in the Introduction, an intuitive grasp can be obtained by thinking of a blackboard where output plans are posted. Plan revisions take place until no firm wishes to revise its plan given that other firms’ plans will be enacted. An extensive form representation of the plan revision game is shown in fig. 1. Starting at A with stated plans qi, Q_i, where the choice of i is arbitrary, firm i has to revise its plan without knowing whether other firms will also revise their plans. Thus firm i does not know which node in the information set B is appropriate. Given the information in Q-i3 firm i decides whether to change its plan to qi or leave it as it is. If other firms do not change their plans, then a decision not to change the ith firm’s plan means that each firm is happy to enact its plan given other firms enact theirs. If this is not the case at least one firm will change its plan and the game returns to A with updated plans. It is clear that each tit-m has to act as if it is the firm’s last chance to revise its plan. Thus as the only equilibrium is one where no firm is unhappy with its plan, there is no payoff from procrastination, and each firm will simply set its plan
238
M. Hviid
and N.J. Ireland,
Model
consutent
beliefs with information
transmission
in oligopol)
produce conlrary to plan. Suspend blackboard game
Produce Pi* Q.i proceed to next repetition
Q-i
go back to A with updated information based on new plans
Fig. 1. Output plans in the blackboard game.
at that output, given all its information embodied in Q_i, which will maximise its expected profit. However the information provided by the ith firm’s output plan leads to an equilibrium which departs from the standard Cournot-Nash equilibrium. It is the extra expected profit from this equilibrium compared to the Cournot-Nash equilibrium that allows the firms to be disciplined in terms of keeping to their last announced plans. Deviations are punished by the industry suspending the blackboard game and reverting to Cournot-Nash for a sufficient number of periods to make deviating unprofitable almost certainly. An equivalent punishment to suspending the blackboard game is simply the natural one of the removal of output plan credibility. The information transmission and payoffs from the blackboard game will be investigated below within the framework of a rational expectations equilibrium using the technique known as undetermined coefficients. In order to decide on an output level, a firm must form an expectation concerning how the rest of the industry’s beliefs will be affected and how this will affect other firms’ behaviour. This will depend on the firms’ own information as well as the information and market pre-emption message incorporated in the ith firm’s planned output. Let the conjectural function held by the ith firm be
M.
Hviid
and NJ.
Ireland,
Model
consistent
beliefs
with in/ormnrion
Q_i=a+bS’_i+cqi.
transmission
in oligopoly
239
(12)
where S’_i is the ‘aggregate’ signal (to be defined below) on which the other firms base their choice of strategies. Note that (12) assumes that plans have already been chalked up on the blackboard; initially plans are made given only each firm’s private information. Thus the conjecture is solely on the effect of observable plans. The form of (12) reflects the fact that if firm i observes Q _ i it would be able to infer additional information on a by solving for S’-i. The constancy of the conjecture c is obviously a limitation.’ However, it should be noted that the blackboard game allows full investigation of responses so that local beliefs must accord to global conjectures. Also the symmetry of the model suggests that a natural form of the conjecture should reflect the form of Cournot reaction functions. Due to this structure of the model, the first-order condition for firm j must be linear in Sj and vi. Hence Q_i must be linear in all S, vi, jS i. The signal on a obtained from Q-i can thus be written as: S’_i =
C (Sj-kqj)/(n-
1)~
C SJI(nj#i
j+i
=x+ 1 (Ej_kqj)/(n_l)=a+8i,
1)
(13) (13’)
j#i
where
SJ=Sj-k~j~
Bi=
C
(Ej-kqj)/(n-
1)
j#i
and k is a parameter which reflects the relative importance of Sj and 9j on other firms’ outputs. Observing Q-i only partially reveals the information held on a by the other (n- 1) firms (the Sj) due to the ‘noise’ arising from random costs (qj); only the S; are inferred. As firms have to announce their output plans prior to carrying them out, the quantity Q-i is available to the ith firm. From its conjectural function (12), S’-i can then be inferred. Thus any candidate for an equilibrium set of output plans ql, q 2,. . . ,qn must incorporate the information contained in S’_1, s- 2,. . . , S’_, respectively. Each firm i has two signals, Si and S’_i to use for predicting the demand parameter a. Firm i’s information on a given in (2) and (13’) can be summarised, see Jaffe and Winkler (1976, p. 50, 57-58), as
‘See Boyer and Moreaux (1983a. b).
240
M. Hviid and N.J. Ireland, Model consistent beliefs with information transmission in oligopoly
Si=6Si +( 1_6)S’_i,
(14)
VAR(Si)=of
(19
+sof,
where 6 = VAR(ei)/[ VAR(Ei) + VAR(ei)] =(e,Z + k2ei)/(qef + k2ai)
(lo)
E(~i1Si,rli)=[(tSi+(l_t)di-_i_Q)qi]
(17)
and
subject to (12). Note that (5) is now the special case in which 6= 1, i.e., the case where either crf 3 0, in which case Si contains sufficient information on a or CJ:+ cc in which case S’_i is an infinitely noisy signal and thus has no value. 3. Equilibrium
with model consistent expectations
We define an equilibrium with n active firms as follows: Definition 1.
An n-tuple(q,,q,, . . . , q,,) is an equilibrium if, given information (Si,qi) no firm i wishes to change qi, where S’-i=(Q-i-a-cq)/b and (a, b,c) form a consistent conjecture.
In such an equilibrium no firm wishes to change its plan. Of course each firm tries to manipulate the oligopoly process by choosing the strategically most advantageous output plan: however, other firms can interpret such rational behaviour and infer the correct market signals. The complete equilibrium process of the game depicted in fig. 1 incorporating information transmission can thus be summarised in the following. (1) Each tit-m states it best output plan given its conjectural function (12) of competitors’ reactions, and given its private information. (2) Any firm i can revise its output plan given the information S’_i which is extracted from the initial plans.
M. Hviid and NJ.
Ireland,
Model consistent
beliefs with information
transmission
in oligopoly
Z-t1
(3) No further revisions are desired and the plans are put into effect. If any firm ‘pretends’ information at stage 1 or 2 then further iterations are required to attain the same equilibrium. Alternatively the pretending firm has to adopt a sub-optimal plan and realise less expected profit. Thus the above sequence is sub-game perfect. Of course, the simplicity of the outcome is due to the binding nature of final plans. A last-minute change of plan by one firm means that all other firms will still be able to adjust their behaviour: talk is not cheap, This can be compared with Mailath (1989), where the ‘talk’ is replaced by real outputs.* We show in the Appendix that there exist two sets of coefficients a, b,c, such that the conjectures embodied in (12) are consistent with behaviour dictated by maximising (17). When firms all hold common consistent conjectural functions, responses to variations in firm behaviour correspond to that predicted by the conjectural function. Proposition 1 below focuses on the coefficient c( =dQ_,/dq,) in the conjectural function (12), since this is paramount in determining the observed degree of competition in equilibrium. Proposition 1. There are two consistent conjectural functions. In one case the coeficient c= - 1, and this is the competitive conjecture. In the other case c=c* where c*=(2-(n+l)J*)/b* and s*~[l/n,l] is uniquely determined by (13), (4’) and k=(t6)-‘. Proof.
See appendix.
Since the competitive conjecture is inferior in terms of pay-offs, we will concentrate our attention on the c* conjecture.g A number of important properties of the c* conjecture can be stated immediately from Proposition 1. (1) c* is an increasing function of 0,2, a decreasing function of 0, and an ambiguous function of r_rtand n. (2.i) As ~~+O,c*+n-1. (2.ii) As C: + af/[f(n + 1) + o:/o,~], c* + 0. (Ziii) As C: +@,c*+ - 1, where @=[n/(n-2)]az/[f(n+ l)+of/az]*. (3) IfO
242
M. Hciid
and N.J. Ireland,
Model
consistent
belie>
with information
transmission
in oligopoly
The sign of the c* conjecture in Proposition 1 relates to output strategies incorporating information transmission. Thus more uncertainty over the prior predictor of a (higher ai) will lead to a higher value of c* as firms put more reliance on market signals as a whole. Similarly as c,’ increases, indicating less information content in output plans, so firms rely more on their own private signals and the coherence of the market is reduced. The effect of increased variance in the private signal (higher 0:) is two-fold; first it leads to more reliance on market signals relative to private signals, but secondly it leads to more reliance on the prior predictor so that other firms’ outputs become less important for their information content concerning market demand, and relatively more important as indicators of market supply. Similarly, an unambiguous effect from increasing n is not forthcoming. The specific solution 2.i above deserves further comments. In this case it is as if firms simultaneously collude over their choice of quantities (c* = n - 1) and pool their information (6* = l/n). With 0,: close to zero, Q-r becomes a sufficient statistic for Cj~i Sj and the apparent pooling of information follows using Theorem 1 in McKelvey and Page (1986). In this special case market information is imparted fully and involuntarily through announced plans, and the collusion equilibrium is obtained. For some parameter values, expected profits may be small and even negative. In these situations we would expect either some firms to exit the market or that the blackboard game would break down and be replaced by a simple Cournot-Nash supply process, or both. In our discussion below, we ignore these and other complications by assuming that the number of firms are fixed, that the blackboard game is viable and that all firms always produce positive quantities in equilibrium. Some examples of parameter values supporting the discussion are given at the end of this section. Output in equilibrium
Solving for the three parameters in (12) for the c* conjecture case we can write down the reaction function of firm i as: qi = {(&I-
1)&?+(6n-
+[2-(n+l)E]Q_i}/G
l)st(s:-E) i=l,...,
n,
where G=(n- l)a+(n-2)[2-(n+ 1)6]. Tedious manipulations following proposition concerning outputs and expected profits.
(18)
give us the
Proposition 2. In an equilibrium with consistent conjectural variation c*, aggregate output Q, individual output qi and expected profits E(ni) of $rm i before the signals are received, are respectively
M. Huiid and N.J. Ireland,
Model
consistent
beliefs with
information
transmission
in oligopoly
243
(i) Q=+IcG++&Z(SJ-E)
(19)
(ii) 4i=tscr+A(SI-a)+(f6t-A)~(SJ-a)/n,
(20)
where A=6t(6t-l)/[(n-1)(2-6n)]>Ofor
l/n<6<2/n,
(21)
(iii) E(~i)=~{6(2-n~)o72+t~(2-n~t)a,Z-(f6t)za~+[26/(2-n~)-1]a,Z},
(22)
providing all n firms produce at positive output levels.” Proof.
See appendix.
The equilibrium associated with the c* conjecture described in 2 is a function of all the information in the supplying industry. assumed qi >O Vi,E(ITJ >OVi is also implied in the absence of Further intuitive feel for the equilibrium can be obtained by expected values of the outputs and price-cost margins.
Proposition As we have fixed costs. considering
Proposition 3. Ex ante of any signals being receiced, the c* conjecture yields an equilibrium with the following characteristics (i) The expected industry output is E(Q)=fn& and is increasing in n, ti, and and decreasing in of. (ii) The expected output per firm is E(q) = $%i and is increasing in 5 and o: and decreasing in n and of. (iii) The expected price-cost margin is E(p- m) = ji( 1 - +n8) and is increasing in d and oi and decreasing in n and o:.
a,f,
Proof. E(p-m)
E(Q) and E(q) are found by taking expectations
is found by taking expectations of eq. (A.ll) Differentiation yields the comparative static results. 0
of (19) and (20). in the Appendix.
Proposition 3 shows that the average effect of greater c,’ or smaller 0: is to move the industry towards a more coherent structure with associated lower industry supply and higher price-cost margins. Numerical results and enforcement
A full solution to the equilibrium can be found only by numerical methods “To see why the assumption that all n firms produce at a positive level is necessary, note that and qi+co if Si-a>p, a/n+tp/n if S;-z=p, or --co if S;-zcp, where p=Zj(S;-a)/n. Further, from Proposition 1(2.iii)6-+2/nac* + - 1. Thus as we approach the competitive conjecture, the output levels given in (20) are not bounded. As 6+2/n, only the firm with the most encouraging signals will produce and supply the whole market. 6-r2/n=A+cO
244
M. Huiid and NJ.
Ireland, Model consistent
Table The numerical ”
z or z ?n : C*
E(n) E(CN --II) E(Col-fI)
results:
beliefs with information transmission
in oligopoly
1
5= 10 and 0:=4,
2 4 0.01 0.5055 2.978
2 4 0.10 0.5456 2.833
2 4 0.40 0.626 2.597
0.9566 12.83 12.00 12.83
0.6658 12.73 12.03 12.85
0.1944 12.10 12.10 13.08
E(CN-U) is protit in the Coumot-Nash collusion case.
( =8 in col. 5).
0.10 0.5813 1.970
2 4 0.10 0.5330 2.376
0.4407 12.61 11.54 12.84
0.7526 13.25 11.92 13.31
2
1 I
case: E(Col -l7)
5 4 0.10 0.2769 4.612 1.223 4.410 2.927 5.168 that in the
since 6* is only defined implicitly. It is important however to examine how the expected profit from playing the blackboard game exceeds that of simple Cournot-Nash strategies [see (IO)]. Without such a premium there could be no discipline in enforcing firms to produce according to their final announcements, and the blackboard game would have no credibility. The numerical analysis is summarised in table 1. Numerical results comparing the expected profits of the coordination game and the Nash game. The first three columns consider a duopoly with r~f=a: =4. The amount of noise in information transmission is measured by G: and this increases across the three columns. When it is insignificant (0.01) expected profits from the blackboard game approximates the ‘collusion’ expected profits [eq. (1 l)]. When it is 0.40 there is significant noise for the advantage of the blackboard game over a Cournot-Nash game to disappear. Notice that this advantage disappears at a positive conjecture (c*z 0.19). This reflects the fact that the Cournot-Nash equilibrium yields higher expected profits as cost and demand variations increase: the blackboard game lowers this advantage and only performs better than the Cournot-Nash when cr,’ is smaller than the value which would yield c* =O. When 0: =O.lO, over 80% of the loss from Cournot-Nash compared to the full collusion is made up by playing the blackboard game. These examples show how important the noise r~,’ is in determining the degree of monopoly in the equilibrium. If G,’ is large then firms do not extract much information concerning market demand from the blackboard game and so individual firms only have an incentive to pre-empt the market, leading to high outputs and low prices. Columns 4-6 in table 1 consider the effect on the equilibrium of a better signal (smaller crz), more random demand (higher g.‘) or more firms. All cases can be related to the blackboard case in column 2. A better signal reduces expected profits, and more randomness increases profits, due to the convexity of profits in a. The pattern of the blackboard game yielding most of the difference in expected profits from the collusion case relative to the
M. Hviid and NJ. Ireland. Model consistent beliefs with information
transmission
in oligopoly
245
Cournot-Nash case is repeated. When there are more firms, expected profit is of course lower but the conclusion remains. 4. Conclusions An explanation for the cohesiveness of market supply has been advanced which is based on the transmission of information via quantity plans. Firms take account of the way information will be extracted from plans and used by the rest of the industry. Central to the process is the key conjecture c* which is determined by the underlying parameters of the model (see Proposition 1). It increases as a firm’s prior estimate becomes less reliable (af increases) and as information transmission becomes less noisy (r$ decreases). Thus as the information content of quantity signals become more needed and more efficient, so the industry becomes more collusive in character, and (from Proposition 3) expected industry output decreases and expected pricecost margins increase. Although other contexts have been suggested, an interesting view of our model of a blackboard game is found within a supergame. Because, in each repetition, firms receive some private information about the current realization of the random variables, it is not desirable to specify output for future periods. Imagine that at the beginning of each period firms can meet and talk. Private information cannot be transmitted by direct communication of market demand predictions because truthful revelation of such private information cannot be verified. However, if outputs are observable,” output plans could be made a credible means of information transmission, by punishing deviations from the set of plans to which firms are committed. This transmission would typically not be perfect, but the better the extraction of information (the smaller the ‘noise’ of lirm-specific variations in costs), the more cohesive the industry’s apparent organization and the nearer is price to the monopoly price. Only when there is no cost uncertainty can the monopoly outcome be implemented. In short, the bounds on implicit monopolisation of the industry are set by both the means and the efficiency of market information transmission. Many assumptions, such as linear demand, homogeneous products and constant costs, contributed to the tractibility of the analysis. The central limitation however has been that parameter values were such as to provide a profit premium from playing the blackboard game, and hence provide the mechanism for enforcing the rule of the game: that only current announced plans can be put into effect. The numerical results in table 1 showed examples of acceptable parameter values. Of course, even if the gains from “The case when outputs Abreu, Pearce and Stachetti
are not observable (1985).
is considered
by Green
and
Porter
(1984) and
246
M. Hviid and N.J. Ireland, Mode/ consistem beliefs with information rransmission in oligopol)
retaining for future periods the coordination implicit in the blackboard game are large, breakdown might still be observed. This would be caused by extreme high expected profits in the current period outweighing the gain from conforming to the rule. Such a possibility cannot be excluded since r, ~~ do not have a finite upper bound on their supports, although it would become more remote as the discount factor on the future declined. That the coordination and information transmission may break down in extreme circumstances probably reflect a general feature of games of this kind. The model outlined in this paper could be extended to relax a number of assumptions in order to apply the underlying thesis to a wide selection of problems. Possible examples might include an extension to monopolistic competition where conjectural variations are a determinant of whether too many or too few products are produced in a Chamberlinian equilibrium [see Koenker and Perry (1981) and Ireland (1983)]. Further extensions relate to the inclusion of risk averseness of firms, where information transmission would have the secondary effect of reducing risk. It is hoped to explore these extensions in future research.
Appendix Proof of Proposition I. Writing Q = Q _i+ qi, the first order condition for maximising (17) with respect to qi given the conjectural function (12) the combined signal (14) and pi, is
Substituting S’_i from (12) into (A.l) and solving for qi in terms of Si, rli, Q-i and Cryields the reaction function, incorporating reaction to information as q,=(l-t)!X&zar(l-@+ 1 2+c+cr(l-6) where S:=S,-(
tab
2+c+ct(l-6)
S!+ t(l--4-b ’ 2+c+ct(1-6)
Q
ir
(‘4.2)
l/tS)rli implying, using (13’) together with (4’) that
(A.3)
Note that a unique value of k greater than one exists to solve (A.3). Using (16) and (4’), the right-hand side can be written as
M. Huiid and NJ.
not +
Ireland,
Model
consistent
belitfs with information transmission
in oligopoly
247
kza2 a2 “+A.
af
a,2+k2a,2
This is a decreasing function of k, and for all k, strictly greater than one. Further it is bounded at 1 +~:/a: as k -+ m. Therefore (A.3) has a unique positive solution and we will simply denote this as k. A reaction function of the form (A.2) holds for any ith firm. We must now investigate the joint response of all other firms, given their reaction functions (A.2), to an increase in one firm’s output. Following Perry (1982) we not only take into account the direct effects but also the interaction among all the other firms (if firm i and i react to a change in the output of firm 1, then they also react to each other’s reaction). Without loss of generality we focus on firm 1. For i# 1, we can write (A.2) as t(l-6)-b %- 2+c+ct(l-6) t6b
+2+c+ct(l-6)
Q__ _(l-t)bE--at(l-6) ‘.t2+c+ct(l-6) p+ t(l--4-b I 2+c+ct(l-6)
41y
(A.4)
where Q-i.1 is the sum of outputs of all but the ith and the first firms. Summing (A.4) over i from 2 to n yields
t6b +2+C+Ct(l-6)
i S;+(n_l) i=2
r(l-4-b 2+c+ct(l-6)q1*
(A.9
The actual response function (AS) has the same form as the conjectured response function (12). If the values of a, b and c in (12) can be found such that (12) and (A.5) are identical, a consistent conjectural variations equilibrium has been identified. Equating the coefficients in (12) and (A.5) we get [(l-t)bl-r(l-6)a](n-1) ‘=b(2+c)+ct(l-6)-(n-2)(r(1-6)-b)’ b=
(n- l)t6b b(2+c)+ct(l-6)-(n-2)(t(l-6)-b)’
(A-6)
(A.7)
248
M. Haiid and N.J. Irelund, Model consistent beliefi with information
transmission
in oligopoly
(n-l)(t(l-@--b) C=b(2+c)+ct(l-6)-(n-2)(t(l-S)-b)~
(‘4-Q
Take the ratio of (A.7) to (A.8) and solve to get b=t(l-6)-t&.
(A.9)
Insert this in (A.8) and rearrange to get sc2--[2-(n+2)6]c-[2-(n+1)6]=0.
(A. 10)
The two roots of (A.lO) are the conjectures given in the proposition. From (A.9) and (A.@ we see that for each of the two values of c, there is a unique value of a and b. 0 Proof of Proposition 2. Part (i) and (ii) are found by solving (18). To show (iii), note that the price-cost margin for the ith firm is cc--Q-vi. Using Q from (i) above a, q and qi
allows it to be written
p-m--?fi=a-+l&!-+&
i
purely
in terms of the random
variables
(s;-di)-~i.
(A.1 1)
i=l
Combine (ii) and (A.1 1) and take unconditional
+6cr+A(S;--q+
x [
where A is defined (iii).
in
the
expectations
to get
1 1j$l II )&-A
proposition.
;
Solving
(SJ-6)
and
,
rearranging
yields
0
References Abreu, D., Journal Boyer, M. Journal Boyer, M. theory:
D. Pearce and E. Stachetti, 1985, Optimal cartel equilibria with imperfect monitoring, of Economic Theory 39, 25 l-269. and M. Moreaux, 1983a. Conjectures, rationality and duopoly theory, International of Industrial Organization 1, 2341. and M. Moreaux, 1983b. Consistent versus non-consistent conjectures in duopoly some examples, Journal of Industrial Economics 32, 97-l 10.
M. Hviid
and ‘V.J. Ireland,
Model
consistent
beliefs with information
transmission
in oligopoly
249
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