Aggregation of Information in Large Cournot Markets
Author(s): Xavier Vives
Source: Econometrica, Vol. 56, No. 4, (Jul., 1988), pp. 851-876
Published by: The Econometric Society
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Econometrica,Vol. 56, No. 4 (July, 1988), 851-876
AGGREGATION OF INFORMATION IN LARGE
COURNOT MARKETS
BY XAVIERVIVES1
Consider a homogeneous product market where firms have private information about an
uncertain demand parameter and compete in quantities. We examine the convergence
properties of Bayesian-Cournot equilibria as the economy is replicated and conclude that
large Cournot (or almost competitive) markets do not aggregate information efficiently
except possibly when the production technology exhibits constant returns to scale. Even in
a competitive market there is a welfare loss with respect to the first best outcome due to
incomplete information in general. Nevertheless a competitive market is efficient, taking as
given the decentralized private information structure of the economy. Endogenous (and
costly) information acquisition is examined and seen to imply that the market outcome
always falls short of the first best level with decreasing returns to scale. The results are also
shown to be robust to the addition of extra rounds of competition which allows firms to use
the information revealed by past prices. Explicit closed form solutions yielding comparative
static results are obtained for models characterized by quadratic payoffs and affine
information structures.
KEYwoRDs: Information aggregation, Cournot limit, Bayesian equilibrium, competitive
market, rational expectations, uncertainty, information acquisition.
1. INTRODUCTION
MARKETS
Do COMPETITIVE
aggregateinformationefficiently?This question has
been answeredin the affirmativeby Wilson(1977) and Milgrom(1979 and 1981)
in an auction context and more recentlyby Palfrey(1985) in a Cournotmodel.
Wilson (1977) considersa first-priceauctionin which buyershave some private
informationabout the commonvalue of the good to be sold and shows (under
certain regularityconditions)that as the numberof biddersgoes to infinitythe
maximumbid is almostsurelyequal to the true value. Milgrom(1979) obtainsa
necessaryand sufficientinformationconditionfor convergencein probabilityto
the true value. The Vickreyauctionhas the same type of limitingpropertiesalso
(Milgrom (1981)). Palfrey (1985) considers a Cournot market with constant
marginalcosts wherefirmshaveprivateinformationabout the uncertaindemand
but are otherwiseidentical.He shows, under mild regularityconditionson the
informationstructure,that the randommarketprice convergesalmost surelyto
the full informationcompetitiveprice as the numberof firmsgoes to infinity.A
similarresulthas been obtainedalso by Li (1985).Accordingto these analyses,in
the limit competitivemarketthe full informationfirstbest outcomeprevailseven
though no firm knows the true state of demand.
All these convergencepropertieshold in a context where information is
exogenous and agents (biddersor firms)do not observe any aggregatemarket
statistic before makingtheir decisions;they can only conditionon their private
1I am grateful to Roger Guesnerie, Ken Judd, Rich Kihlstrom, Paul Milgrom, Tom Palfrey, and
Rafael Rob for helpful comments. The suggestions of two anonymous referees helped to improve the
exposition of the paper substantially. Byoung Jun provided excellent research assistance. Financial
support from NSF Grant IST-8519672 is gratefully acknowledged.
851
852
XAVIER VIVES
information.The resultsobtainedwhenthe marketmechanismis an auctionor of
the Cournot type suggest, as Palfrey puts it, "that in large enough markets,
diverse privateinformationdoes not drasticallyalter the conclusionreachedby
the idealized Walrasianmodel of pricingin a full informationworld"(Palfrey
(1985, p. 80)). In otherwordsthereshouldbe no welfareloss with respectto the
full informationfirstbest in competitivemarketswith incompleteinformation.
The purposeof this paperis to examinethe aboveconjecturein the contextof
the Coumot model. We will examine the asymptoticpropertiesof BayesianCournot equilibriaunder differentassumptionsabout technology,the information acquisitionprocess,and marketobservables.Two main conclusionsemerge
from our analysis(withthe possibleexceptionof constantreturnsenvironments):
(a) thereis a welfareloss with respectto the firstbest in largeCournot(or almost
competitive)marketswith incompleteinformation,and (b) a competitivemarket
is secondbest efficientwhentakingas giventhe decentralizedprivateinformation
structureof the economy.
Conclusion (a) is based on a very simple idea. Considera large Cournotor
almost competitivemarketwherefirmsreceiveprivate(and costless)noisy signals
about the uncertain demand and have identical decreasingreturns to scale
technologies.A necessarycondition for full informationfirst best efficiencyis
that conditionalon the true state of demandidenticalfirmsproduceat the same
marginalcost, but this is incompatiblewith firmshaving differentappraisalsof
demand when marginalcosts are increasing.In purely statistical terms there
simply do not exist decentralizedproductionstrategies(equilibriumor not) which
yield asymptoticallyefficientbehavior.Therefore,there is in general a welfare
loss due to incompleteinformationin a competitivemarketexcept if marginal
costs are constant. With constantreturnsto scale only total output mattersfor
welfarepurposesand, againin purelystatisticalterms,a necessaryconditionfor
the marketto be efficientis that thereexist an unbiasedestimatorof the firstbest
supply. Palfrey and Li's regularityassumptionson the informationstructuredo
imply the existenceof such an estimator.
In general, then, a large Coumot market will not aggregateinformation
efficientlyalthoughif firmswere to pool theirinformationthe aggregatestatistic
would be a perfect signal and we would obtain the full informationWalrasian
outcome.2 Conclusion (b) makes the point that if we insist on decentralized
decision makingthe marketdoes as well as possible.More precisely,the market
works like a team wheredecentralizedproductionrules are assignedto firmsto
maximizeexpected total surplus.This result follows easily from Radner'swork
on team theory(Radner(1962)).
Consideringmodelswith liner demand,quadraticcosts, and affineinformation
structurewe are able to obtainclosed-formsolutionsand to computethe welfare
loss explicitly,showingthat it is decreasingwith the precisionof the information
of the firmsand increasingwith the basic uncertaintyof demand.
2The incentivesfor informationsharingin oligopolymodelswith uncertaindemandhave been
analyzedby Novshek and Sonnenschein(1982),Clarke(1983),Vives (1984),Gal-Or(1985),and Li
(1985) amongothers.
AGGREGATION OF INFORMATION
853
The inefficiencyof large Cournotmarketswith private informationis reinforced when consideringendogenousacquisitionof information.3In this case the
firstbest outcomewith decreasingreturnsto scalecan only be attainedif the cost
of informationis zero. Otherwisethereis alwaysa welfareloss. Nevertheless,as
before, a planner could not improve upon the market if only decentralized
strategiesare feasible.In a competitivemarketwith negligibleagents a firmhas
the right (second best) incentives to acquire information.We show that the
equilibriumprecision of informationpurchasedis declining in its cost and
increasingin the priorvariabilityof the uncertaindemand.Withconstantreturns
to scale the analysisis more delicate.No equilibriumexists with costly information acquisitionin a purelycompetitivemarket.Firmscannotmakeany profitsat
the market stage contingenton the informationthey receive (if they did they
would expand output indefinitely).Thereforeno equilibriumwith positive researchis possible.But if no firmdoes researchthen thereare enormousincentives
for a firm to buy informationand consequentlyno equilibriumexists. Nevertheless this argumentis no longer valid if firms retain some (maybe very small)
monopoly power. In fact we can show that by replicatingappropriatelythe
market,so that monopolypowervanishes,equilibriumwith endogenousinformation acquisitionexists and approachesthe firstbest outcome.
We may wonder whetherthe inefficiencyarises because we have considered
only one-shot competitionin the marketand thereforewe have not given firms
the opportunityto learn about demandconditionsfrom prices. We claim that
this is not the case: ourresultscan be shownto be robustto the additionof extra
rounds of marketcompetitionin which the informationrevealedby past prices
can be used by firms.Althoughprices will be fully revealingin the competitive
economy,in the interimperiod(firmscan learnonly from past prices)therewill
be a welfareloss and the incentivesto purchaseinformationwill be unchanged.
2. A METHODOLOGICALNOTE AND OUTLINE OF THE PAPER
It is our contention that the appropriatemethod to analyzethe convergence
propertiesof the Cournotmechanismunder incompleteinformationis to consider replicatedmarketswhere demandis replicatedat the same time that the
number of firms increasessince in this mannera well definedcompetitivelimit
market is obtained. This methodologyis in line with the literatureon large
Cournotmarketswith completeinformation(see Novshek (1980)), and with the
view that the continuummodelis the appropriateformalizationof a competitive
economy(see Aumann(1964)).In the courseof our analysiswe will highlightthe
importance of the replica assumptions.Furthermore,reasoning in the limit
economy it is very easy to understandthe statisticalreasonswhy a competitive
marketwith incompleteinformationwill not aggregateinformationefficientlyin
generaland to characterizethe equilibriumand its secondbest properties.This is
what is done in Section3.
3It has already been noted that costly informationacquisitionmay change the convergence
propertiesof the marketmechanismbe it auction-type(Matthews(1984)) or Cournot-type(Li,
McKelvey,and Page (1985)).
854
XAVIER VIVES
The rest of the paper considers models with quadraticpayoffs and affine
informationstructurewhich yield computableBayesian-Cournotequilibriaand
parameterizethe precisionof the informationof firms.In Section4 we check in
the context of this model that the Bayesian(competitive)equilibriumof the limit
economy is not an artifact of our continuum specificationbut the limit of
Bayesian-Cournotequilibriaof the replica markets. Furthermoreclosed-form
solutions are derivedfor the competitiveequilibriumand the associatedwelfare
loss and comparativestaticsanalysiswith respectto parametersin the information structureis performed.
Section 5 deals with endogenous information acquisition. We consider a
two-stagegame wherefirstfirmsdecide how much (precisionof) informationto
purchaseand then competein the marketcontingenton the chosen precisions.
Again closed-formsolutionsare obtainedfor the equilibriumin the continuum
economy(with decreasingreturnsto scale)and it is checkedthat this equilibrium
is the limit of a sequenceof unique subgameperfect equilibriaof the replica
markets.Comparativestaticsand welfareanalysisof the competitiveequilibrium
follow easily.
In Section 6 we examine the robustnessof the above results to a situation
where firmscan learnfrom past prices.A thirdstage wherefirmscompeteagain
after having observedthe second stage marketprice is added to the model of
Section 5. Concludingremarksfollow.
3. AN HEURISTIC ARGUMENT IN A COMPETITIVE MARKET
Considera competitivemarketwith downwardsloping inversedemandgiven
by p = P(x; 0) wherex is per capita output and 9 a randomparameter.Firm i
receivesa noisy estimateof 9, si. The signalsreceivedby firmsare independently
and identicallydistributedgiven 9 and, withoutloss of generality,are unbiased,
E(si IH)= 0. Thereis a continuumof firmsindexedby i E [0,1], each one of them
with strictlyconvex twice-continuouslydifferentiablevariablecosts given by the
function C(.). There are no fixed costs. We will make the conventionthat the
Strong Law of Large Numbers(S.L.L.N.)holds for our continuumeconomy.4
That is, given 9, the averagesignalequals9 almostsurely(a.s.).The argumentsin
this section shouldbe viewedas heuristic;in the other sectionswe will deal with
the continuumcase as a limit of finiteeconomies.
In this situation a productionstrategyfor firm i is a function xi(-) which
associates an output to the signalreceived.Supposethus that we have a market
equilibriumcharacterizedby xi( *), i E [0,1]; that is, xi( *), i E [0,1], is a Bayesian
Nash equilibriumof our continuumeconomy.5xi(si) must maximizeE(nils) =
xiE(P(x; O)si) - C(xi) wherex is averageoutput, x = Jxi(si)di. Note that the
4Note that we have a continuum of iid random variables according to our specification. Judd
(1985) has shown that this assumption can be made consistent with measure theory.
5See Schmeidler (1973) for a formulation of Nash equilibrium in nonatomic games with finite
strategy spaces and Mas-Colell (1986) for a reformalization of the model in terms of distributions
with an application to games of incomplete information.
AGGREGATION OF INFORMATION
855
equilibriummust be symmetric,xi(si) = x(si) for all i, since the cost functionis
strictly convex and identicalfor all firmsand signals are iid (given 0). Consequently the randomvariablesx(si) will be iid (given 0) and their averagex(#)
will be nonrandom and equal6 to E(x(si)10) accordingto the S.L.L.N. An
interior equilibriumx(.) (that is, one in which x(si) > 0 a.s.) is thus characterized by E(P(x;
)js1) = C'(x(si)); given that firm i has received signal si
the expectedmarketpricemust equalmarginalproductioncosts.
How does the marketoutcome comparewith the full informationfirst best
where total surplus(per capita)is maximizedcontingenton the true value of 0?
Given 0 total surplus (per capita) with averageproduction x is TS(x; 0) =
foxP(z; 0) dz - C(x) and, obviously, first best production x0(0) is given by the
unique x which solves P(x; 0) = C'(x). If firmswere able to pool their private
signals they could conditiontheirproductionto the averagesignal,which equals
0 a.s., and attain the first best by producingx0(0). Will a competitivemarket,
where each firm can condition its productiononly on its private information,
replicatethe firstbest outcome?
A necessaryconditionfor any (symmetric)productionstrategyx(.) to be first
best optimal is that, conditional on 0, identical firms produce at the same
marginal cost, namely, C'(x(si)) = C'(x0(0)) a.s. but with increasing marginal
costs this can happen only if x(si) = x0(0) a.s., which is basically the perfect
information case. Thereforewe should expect a welfare loss in a competitive
marketwith noisy signalssince a productionstrategywhich is not based on the
averagesignal cannot attain the first best. Proposition1 below states the result
and a more formalproof follows.
PROPOSITION 1: In a competitive market with symmetric firms with strictly
convex costs which receiveprivate noisy signals about an uncertaindemandparameter, there is a welfare loss with respect to the full informationfirst best.
PROOF: We show that no symmetricproductionstrategyx(-), and thereforein
particular no competitiveproduction strategy, can attain the first best. The
marketexpected total surplus(per capita)contingenton 0 is given by
E(TSI0) = J(P(z;
0) dz - E(C(x(si))l0).
This is strictly less than the first best TS(x0(0); 0) since TS(x0(0); 0) >
TS(R(0); 0) > E(TSIO).The firstinequalitybeing true since xo is the firstbest,
the second since the cost function is strictly convex, Z(0)=E(x(sj)j0), the
signals are noisy (which means that given 0, x(si) is still random) and in
consequence C(&x(0))< E(C(x(si)) I0).
Q.E.D.
In this context and in general a necessaryconditionfor the market outcome to be
first best optimal is that marginal costs be constant. Firms will produce then at the
6Equality of random variables has to be understood to hold almost surely. We will not insist on
this always.
856
XAVIERVIVES
same marginalcost by assumption.This is the situation consideredby Palfrey
(1985). He finds that if the informationstructureis "regularenough"first best
efficiencyis achievedin the competitivelimit. The intuitionof the resultis simple
enough. Suppose that in the above describedcontext marginalcosts are zero
(without loss of generality)and that inversedemandintersectsthe axis. Firm i
will maximizeE(-rjIs1)= xjE(P(2; 0)jsi) where,as before, x is averageoutput.
Obviouslyin this constantreturnsto scale context a necessaryconditionfor an
interiorequilibriumto exist is that E(P(x&;0) Isi) = 0 for almost all si. Looking
at symmetricequilibria,xi(si) = x(si) for all i, we know that averageproduction
given 0 is nonrandom,x (0) and thereforeE(P(x; 0)I0) = P((x; 0)I0). For first
best efficiency we need: E(P(xZ(0),0)si) = 0 for almost all si to imply that
P(x&(0),0) = 0 for almostall 0.
Palfrey'sresult states that if the signaland parameterspacesare finite and the
likelihood matrix is of full rank (and demand satisfies some mild regularity
conditions),then symmetricinteriorcompetitiveequilibriaare firstbest optimal.
This is easily understoodwith a two-point supportexample: 0 may take two
values, 0 or 0, 0 < <s #, with equal priorprobability.Firm i may receivea low
(s) or a high (s&)signal about 0 with likelihood Pr(sI0) = Pr(sI0) = 1, where
2 < 1 < 1. If 1= 2 the signal is uninformative; if 1= 1 it is perfectly informative.
Let p- and p be respectivelythe equilibriumpriceswhen demandis high (0) and
low (0). Then
LE(p
I
1
rE(pis)
)]
L[
1 -lp
I
0o
A][LP]
Lo]
If the likelihood matrix is of full rank, i.e., if 1> ' then p=p=0 and the
symmetricinteriorequilibriumis efficient.
In the constantreturnscase in orderfor the firstbest optimumto be achieved
it is only necessarythat total output (conditionalon 0) be x0(0). Therefore,a
necessaryconditionfor (Bayesian)competitiveequilibriato achievethe firstbest
is that thereexist some nonnegativevaluedfunctionx(-) such that E(x(si) 0) =
x0(0) for all 0. That is, the functionx(.), the firm'soutput strategy,must be an
unbiased estimateof the firstbest supplyx0(0). In this case x(.) is a Bayesian
competitiveequilibriumsince P(x0(0), 0) =0 and Z(0)= E(x(si)IO). Palfrey's
conditions again imply that this functionexists.
We can view the above results in a purely statisticalway. With decreasing
returnsto scale thereare no decentralized(production)strategieswhichyield an
efficientoutcomeif signalsare noisy and thereforea competitivemarketwill not
attain the full informationfirstbest. With constantreturnsto scale, if theredoes
not exist an unbiasedestimatorof the first best supply then again there are no
decentralizedstrategieswhichyield efficiencyand a competitivemarketwill fail
to attain the firstbest.7
7It remainsan open questionto characterize
in generalthe class of informationstructureswhich
insure the existenceof suchan unbiasedestimator.
857
AGGREGATION OF INFORMATION
Is,the outcomeof the competitivemarketefficientin some sense in general?In
fact it will be efficientgiventhe privateinformationstructureof the economy.We
show below that the competitivemarketworkslike a team whereeach member
follows the decisionrulex(.) that maximizesexpectedtotal surpluswhen thereis
private information.That is, the marketsolves the team problemwith expected
total surplusas objectivefunction.The team problemwith n firmsis to find n
decision rules xi(.) whichmaximize
E{TS(X; 0))
=
E
P(;
)d
C(x()
1=1
where X=
i7xj (sj). (Here X and Z denote total output.) Radner has shown
of the team
that,-undersome regularityconditions(concavityand differentiability
function among them),the decisionrules(xl,..., xn) are globallyoptimalif and
only if they are person-by-personoptimal, that is if (xi,..., x ) cannot be
improvedby changingxi for some i alone (Radner(1962, Theorem1)). In our
case TS is concavein the outputof the firmssince demandis downwardsloping
and costs are strictlyconvex.The optimaldecisionrules will be determinedthen
by E((dTS/dxi)lsi) = O which is equivalent to the competitive condition
E(P(X; 0)1si) = C'(xi(si)), i= 1,..., n, that we have obtained before. Proposition 2 states the result.
PROPOSITION 2: A competitive market works like a team which chooses decentralized production rules for the firms so as to maximize expected total surplus.
4. QUADRATIC COSTS AND AFFINE INFORMATION
STRUCTURE
Considera marketwith inversedemandgivenby p = 0 - /xi, where/ > 0, x is
per capita output,and 0 a possiblyrandomintercept.Firm i has a quadraticcost
function C(xi) = mxi + xix, where m is possibly random and X is a nonnegative
constant.In our modelonly the level of 0- m mattersand thereforewithoutloss
of generalitywe will considerpricesnet of m.
Suppose now that 0 is distributedaccordingto a prior density with finite
variance a 2 and mean M.Firm i receives a signal si such that si = 0 + ei where ei
is a noise term with zero mean, variance vi and with cov(0, ei) = 0. 1/vi
represents the precision of the signal. The signals received by the firms are
independentconditionalon 0 and furthermorewe assumethat E(Olsi) is affine
in si. These assumptions imply that E(0lsi) = (1 - ti)M+ tisi, where ti = a2/(a2
+ vi) and E(sjlsi) = E(0lsi), cov(si, sj) = cov(si, 0) = a2 for all j # i and all i.
(See Ericson(1969).)Signalsi is moreprecisethan signal s' if vi is less than v,
in which case the mean squaredpredictionerror, E{ 0 - E(0 Isi)}2, is smaller.
(See Vives (1984).) Sometimes we will refer to ti as the precision of the
information,keeping a2 implicitlyfixed.In this case as vi rangesfromO to oo, t1
ranges from 1 to 0.
The typical exampleof an affineinformationstructureis for the priorand the
likelihood densities to be Normal. si may be the averageof a fixed numberof
858
XAVIER VIVES
independentobservationsfrom a Normaldistributionwith mean 0 and variable
varianceor the averageof a variablenumberof observationsni from a Normal
distributionwith fixedvariance.The precisionof the informationis proportional
to the numberof observationsni. Otherexampleswhere the sample mean is a
sufficientstatisticfor 0 and an affineinformationstructureobtainsincludecases
in which the observationsare conditionallyindependentBinomial, Negative
Binomial, Poisson, Gamma or Exponentialwhen assigned natural conjugate
priors. (See DeGroot (1970).) For example, the pairs Beta-Binomialand
Gamma-Poissonwork. In these examples,as well as in the Normal case, s is
more precise(weakly)than s'(v < v') if and only if s is moreinformativethan s'
in the Blackwellsense since a more precisesignalmeans a largersample.
We are interestedin large marketswith many firms.We will study the n-firm
case and its limit as n goes to infinity.It will be convenientto argueagainin the
limit competitivecase first and then check that the competitiveequilibriumis
indeed the limit of a uniquesequenceof Bayesian-Cournot
equilibriaof the finite
economies.
Equilibrium
Thereis a continuumof firmsindexedby i, i E [0,1]. Firm i receivesa signal s1
about the uncertaindemandintercept0. The strategyof firm i is thus a function
xi(-) which associatesan outputto the signalreceived.We will look for Bayesian
Nash equilibriaof this market.We assumethat the signalsreceivedby the firms
are identicallydistributedconditionalon 0. In this case vi= v and ti= t for all i.
Conditionalon havingreceivedsi firm i will maximizeE(jilsi) = E( p si)xi Xxi, wherep = 0 - /3R and 5 is averageproduction,x = fxi(si) di. Given 5 the
optimal responseof firm i is xi(si) = E(plsi)/2X providedX > 0, whichwill be
the same for all firmsand thereforeequilibriawill be symmetric.We have then
that xi = x for all i and E(ZIsi) = E(x(sj)Isi), j 0 i. The first order condition
yields 2Xx(si) = E(O si) - f3E(x(sj)Isi) and it is easily checked that the unique
affine function x(@) which satisfies it is given by x(si) = a(si - ,u) + b,u where
a = t/(2X + fit) and b = 1/(2X + /3). Average output given 0, x(0), equals then
a(O - ,u)+ b,u since the averagesignalconditionalon 0 equals 0 a.s.
When firms receive no information (t = 0), then a = 0 and production is
constant at the level b,u;when firms receive perfect information(t = 1), then
x(si) = asi. As t varies from 0 to 1 the slope of the equilibriumstrategy, a,
increasesfrom 0 to b (see Figure 1). When X= 0 the equilibriumrequiresthat
E( p Isi) = 0; restricting attention to symmetric equilibria, we get that x(si)
=
silB
provided that t > 0; that is, the above formulais valid letting X= 0, since then
a = b = 1/13. If t = 0 and the signals are uninformative, then x(si) = M//. Note
that in the constantreturnsto scale case when the signalsare informative(t > 0),
the equilibrium strategies x(si) = sil/f do not depend on t, the precision of the
information.
The equilibriumwe have described is not an artifact of our continuum
specification;it is the limit of a uniquesequenceof Bayesian-Cournot
equilibria
859
AGGREGATION OF INFORMATION
bp.
Si
FIGURE 1.-Equilibrium strategy of firm i.
as we replicatethe numberof firmsand consumersin the market.The n-replica
market consists of n-firmsfacing an inverse demand given by p = 0 - fiX/n
where X/n is per capitaoutput.Proposition3 states the result.
PROPOSITION 3: Given X> 0 and tE [0,1] the equilibrium of the continuum
economy wherefirm i produces according to x(si) = a(si - I) + b,i with a = t/(2X
+ 18t)(a = 0 if X= t = 0) and b = 1/(2X + /8) is the limit of a unique sequence of
Bayesian-Cournot equilibria of the n-replica markets, xn(si) = an(si - A) + bn,
where an, -* a, bnn-- b and the average output Exn(si)/n given 0 converges almost
surely to x.(O)= a(O-It) + b,i.
PROOF: According to Lemma 1 in the Appendix and letting ti= t for all i
there is a unique and symmetricBayesian-Cournot
equilibriumof the n-replica
market given by
an=
xn(S1)
(2
D (t) =
=
+X)
an(si
-
4( - + A)
-
,)
+ bnu where
) /Dn(t)
+ 2(
+ X)(n
bn= (-+
-2)-t-(-)
2X)/Dn(1)
and
t 2(n -1)
Therefore as n goes to infinity an -4 a and bn -4 b. Almost sure convergence
follows from the StrongLaw of LargeNumberssince,given 0, the signalsare iid
and E(si 0) = 0. Therefore, given 0, &n= ?is1/n converges almost surely to 0
and consequentlyYiXn(si)/n = an(Sn- I) + bnli converges a.s. to a( - It) + b,u.
Q.E.D.
860
XAVIER VIVES
bp
FIGURE2.-Average production.
Welfare
Per capita total surpluswith all firmsproducingthe output x is given by
(2
)
Full informationfirst best productionis thereforex0(G)= bO.Averageproduc= a( - It) + b,i whichis equal
tion in the competitiveeconomy(given0) is xZ(@)
to the full informationfirstbest productionx0(O) only if t = 1 (perfectinformation) or if X = 0 and t > 0 (constantreturnsto scale).Recallthat as t goes from0
to 1 a increasesfrom0 to b. For t < 1 and X> 0, as it is obviousfrom Figure2,
if 0 is high the marketunderproducesand if 0 is low the marketoverproduces
with respectto the full informationfirstbest. In the constantreturnsto scalecase
averageproductionx(O) is independentof the precisionof the informationt and
contingenton 0 the marketproducesthe rightamount.
Expected total surplusat the first best is ETS0 = EG2/2. Market expected
total surplus ETS equals (au2 + b,u2)/2 and thereforethe welfare loss equals
(b - a)a2/2. The market ETS can be computed as the sum of per capita
consumersurplusf3E(x(G))2/2 and per capita(or firm)profitsEsTi=
=E(x(Si))2,
which equal X(a 2Var(si)+ b21u2).Table I gives the principalmagnitudesof the
competitiveeconomy.
The welfareloss with respectto the full informationfirstbest is positivewhen
informationis not perfectexcept if marginalcosts are constant.In the previous
section we have claimedthat a competitivemarketwould solve a team problem
with privateinformationand expectedtotal surplusas objectivefunction.Lemma
AGGREGATION OF INFORMATION
861
TABLE I
x(s,) = a(s, - ,t) + bt,
Es7i= k(a2Var(s,) + b2p3)
ECS = ,8(a2a2 + b2p,2)/2
+ b,2
ETS= -(aa
x(8) = a(O - t,) + bt,
x0(6) = bO
WL = ?(b - a)a2
Equlibrium strategy of firm i:
Expected profits of firm i:
Expected consumer surplus:
Expected total surplus:
Average production given 0:
Full information first best production:
Welfare loss:
t
wherea=
2A
(=Oif X=t=O)and
1
b= 2A+@
1 in the Appendix shows that the n-replicamarketsolves also a team problem
but using
1
n(n+1
x
j
2n
i j
as objectivefunctioninsteadof total per capita surplus
-Tn
n(
x-2n
x)
A
The differencein both functions is 1/n(TSn-Gn)=f1/2n2EYx 2 (noting that
(?x j)2 = YjxJ2+ yi2 1x xj). Considering square integrable sequences (xi)?' 1
both TSn/n and Gn/n convergeto per capita total surplusin the continuum
economy, TS= 0 x- 2/2 - Xfx dj, as n goes to infinity, since EYxJ /nn
0. A competitive economy is efficient as long
Jx dj implies that (TSn - Gn)/n --+n
as the individualproductiveunits have to rely on theirprivateinformationas we
have claimedin Proposition2 above.
How does ETS change with variationsin the precisionof information,l/v,
and in the basic uncertaintyof demand,a2? One wouldhope that improvements
in the precisionof informationwould increaseETS (reducingthe welfareloss).
We will see that this is indeed the case. Increasingthe basic uncertaintyof
demand increasesETS but ETS?,the first best ETS, increasesby more and the
welfareloss is increasingin a
PROPOSITION 4: (a) With increasing marginal costs (X> 0) the welfare loss
increases with the basic uncertaintyof demand (a 2) and decreases with the precision
of the information (1/v); (b) with constant marginal costs (X = 0) the welfare loss
is zero provided that the signals are informative (v < oo or equivalently t > 0).
Otherwise the welfare loss is increasing in 2.
862
PROOF:
XAVIER VIVES
WL = '(b - a))a2. After some computations we get
aWL
au
1
2~
2tl2X( - t)
1(b2
2).
-(2X +j8t)2j2b
It is positive if and only if b > a. This is the case if X > 0 and t < 1 or if X = 0
and t = 0. If X= 0 and t > 0 then b = a and WL= O. WL= ETS -ETS
decreaseswith l/v for fixed a2: lookingat Table I we see thatby increasingt we
increase ETS since a is increasingin t (recallthat t = a 2/(a2 + v)) and ETS0
Q.E.D.
stays constant.
In case (a), why does ETSincreasewith the precisionof information?First,per
capita expectedconsumersurplus(ECS)increaseswith the precisionof information (t) since consumersurplusis a convex function of averageoutput (CS =
,3 2/2) and increasesin t increasethe slope of x (see Figure2) and makeit more
variable. Secoxnd,expected profits may increase or decreasewith t (see Vives
(1986b) for a complete analysis) but the effect on welfare of the consumers
alwaysdominatesthe profiteffect.On the otherhand increasesin a2 benefitboth
firms and consumersbut have a largereffecton the firstbest than on the market
level.
We have claimed that a welfareloss exists with incompleteinformationand
strictly convex costs and furthermorethat the inefficiencyis inverselyrelatedto
the precision of information.It is not difficultto imaginethat the same will be
true replacingquadraticcosts by capacitylimits;afterall, a capacitylimit can be
interpretedas a very steep marginalcost schedule.This is formallycheckedin a
model with constant marginalcosts and capacity limits with a finite support
informationstructure(see Vives (1986a,Section4)). The qualitativepropertiesof
the capacity model are identicalto the one analyzedin this section.
5. INFORMATIONACQUISITION
Do the inefficienciesthat we found in the case in which the precision of
informationwas givencarryover wheninformationhas to be acquiredat a cost?
The analysisof Mathews(1984),in an auctioncontext,and of Li, McKelveyand
Page (1985), in a Cournot context, support the view that costly information
acquisitionchangesthe convergencepropertiesof the marketmechanism.
We will considerthe quadratic-affine
model of the last sectionand assumethat
firms can acquire additionalobservationsat a constant cost. The precisionof
informationof firm i, l/vi, is proportionalto the sample the firm obtains.8We
will suppose then that the cost to firm i of obtaininginformationwith precision
l/vi is c/vi where c is a positiveconstant.Given our informationstructureit will
be more convenientto workwith ti. Recallingthat ti = a2/(a2 + Vi), for given aJ2
8Li, McKelvey, and Page (1985) model information acquisition similarly. Milgrom (1981) and
Milgrom and Weber (1982) consider information acquisition in auctions.
AGGREGATION OF INFORMATION
863
the cost of obtaininginformationof precisionti will be
c
co( ti ) = --:
ti
a21-t.
to purchase informationof zero precision(ti =0) is costless and to purchase
perfectinformation(ti = 1) is infinitelycostly.We model informationacquisition
as follows. At the first stage firm i purchasesinformationwith precision ti (or
l/vi). At the second stage firms receive their private signals and compete in
quantities and a Bayesian-Cournotequilibriumobtains. The t 's are common
knowledgeat the secondstage(in a competitivemarketthougha firmneeds only
to know the averageprecision).We want to restrictattentionto subgameperfect
equilibriaof the two-stagegame.
Equilibrium
Suppose for a momentthat firmshave alreadypurchasedinformation,firm i
having an informationprecisionof ti. We are in a (competitive)marketwith a
continuum of firms indexed by i E [0,1], and we assume that the first stage
choices of firmsare representableby a measurablefunction t:[0,1] -4 [0,1] where
t(i) = ti E [0,1]. Let t denote the averageinformationprecisionin the market.
Assuming that X> 0 (increasingmarginal costs) we can find a competitive
equilibrium of this (continuum)economy as in Section 4. Let us postulate
xi(si) = ai(si - ,u) + b,i as the equilibrium strategy of firm i. We have then that
the expected value of average production x(x = Jxj(sj) dj) given si is E(Z si) =
j(E(0 Isi) - ,) + b,u where a-= Jajdj. We can use then the first order condition
E(pjsi) = 2Xxi(si), where E(p jsi) = E(Olsi) - fE(x jsi), to solve for the parameters ai and b getting ai = t1/(2X + /3i) and b = 1/(2X + 1B).Expected profits for
firm i, E'7T,will equal XE(xi(si))2 and will be increasingin the precisionof the
information of the firm and decreasingin the averageprecisionin the market
since E(x (si))2 = b212+ tia2/(2X + /t)2.
At the first stage firms have to decide how much informationto purchase.
Given equilibriumexpectedprofitsEs7T
at the secondstage the payoffto firm i of
purchasing precision ti (given a 2) will be Pi(ti, t)
t
ti
(2X + ft)2
2+
1
(2X +
2
f)2
1
= E7T -
C
ti
2
-t,
co(ti) which equals
Note that since we are in a competitivesituation the payoff to firm i only
depends on the averageprecisionof informationin the market(t), which firm i
cannot influence.Furthermore,the marginalbenefitto a firmto acquireinformation will be decreasingin t-,the (average)amountof informationpurchasedby
the other firms. More informationin the marketreducesthe incentivefor any
firm to do research.The intuitive idea is that a firm wants informationto
estimate the market price. When firms have better information(t high) the
marketprice varies less since producersmatch better productiondecisionswith
864
XAVIER VIVES
the changing demand. Consequentlyfor high t an individual firm has less
incentive to acquireinformation.The first order condition (which is sufficient)
will be given by
aPi
at
2
XA
c
1
<
2 (i-ti)2
(2X+/ 3)2
'
with equality if ti > 0. Since the equilibrium will be symmetric we let ti = t= t
and solve for t. We get
fa2
t* = maxp\0,
-A2VX
X + a2
'
which in terms of the precision1/v* is
max{o,
a (2L
)
Xic
In the constantreturnsto scalecase (X = 0) thereis a difficulty:no equilibrium
exists for the two-stagegame if informationacquisitionis costly.9 This fact is
easily understood.Notice firstthat in equilibriumit must be the case that no firm
makes any profit at the marketstage.With constantreturnsto scale, equilibrium
at the second stage implies that the expectedvalue of the marketprice conditional on the signal receivedby any firm be nonpositive.Otherwisethe firm
would expand indefinitely.Thereforefirmscannot purchaseany informationin
equilibriumsince this would imply negativeprofits.Furthermorezero expenditures on researchare not consistentwith equilibriumeither.If no firmpurchases
informationthere are enormousincentivesfor any firm to get some information
and make unboundedprofits under constant returns to scale. With no firm
acquiringinformationthe averageoutput in the marketis nonrandomand any
firm with some informationcould make unboundedprofits by shutting down
operationsif the expectedvalue of the marketprice conditionalon the received
signal is nonpositiveand producingan infiniteamountotherwise.10
The fact that no equilibriumwith endogenousinformationacquisitionexists
underconstantreturnsis neverthelessparticularto the continuummodel.With a
finite number(n) of firmsthe aboveargumentdoes not hold and it is possibleto
show that an equilibriumexists. For n large enough the equilibriuminvolves
positive expenditureson researchwhichare decliningin n, convergingto zero as
n goes to infinity. The key observationis that with a finite number of firms
90n this issue I acknowledgea veryhelpfulconversationwith Paul Milgrom.
l?Notice that the nonexistenceargumentdoes not dependon our quadratic-affine
specification.
Our nonexistenceresult is reminiscentof Wilson's analysis of informationaleconomiesof scale
(Wilson(1974)).Wilsonfindsthat the compoundingof constantreturnsin physicalproductionwith
informationacquisitionyieldsunboundedreturnsin modelswherethe choiceof the decisionvariable
appliesthe sameway to the productionof all unitsof output.Obviously,this impliesnonexistenceof
a competitiveequilibrium.Neverthelessin the model in our paper scale (production)is chosen
optimallyin responseto informationand thereforeit is a "non-example"in Wilson'sterminology.
AGGREGATION OF INFORMATION
865
expectedprofitscan be positivein the marketequilibriumand this way incentives
to acquireinformationare not destroyed.With a large numberof firms,given
that a firm's rivals are all purchasinga little bit of information,the optimal
response of the firm is also to acquirea little information.It is not optimal to
purchasezero since the firmimprovesits expectedprofitswith some information
and it is not optimalto do a lot of researchsince informationis costly and the
other firmshave some information.
The equilibriumwe have computedfor the continuumeconomywith decreasing returnsto scale (t*) is not an artifactof our specification.It is the limit of a
unique sequenceof subgameperfectequilibria(S.P.E.)of the replicaeconomies
consisting of n firms facing an inversedemand p = 0 - 13x/n. In the constant
returnsto scale case no equilibriumexists with costly researchin the continuum
economy but zero averageexpenditureon information(t* = 0) is again the limit
of a unique sequence of S.P.E. of the replica economies. When the cost of
researchis zero, firmsobtain perfectinformation.Proposition5 states the result
and a proof follows.
PROPOSITION
5: In the continuumeconomy, each firm purchasingprecision,
t* = max{O f
rc/X
+a2
}
corresponds to the limit as n tends to infinity of a unique sequence of symmetric
subgame perfect equilibriaof the n-replica markets.
PROOF: Given X 0 and n firms in the market,consider the second stage
wherefirm i receivesa signalof precisionti. FromLemma1 in the Appendixthe
unique Bayesian-Cournotequilibriumof the second stage is given by Xin(Si)=
ain(si-
Au)+ bnAiwhere
n
in
Yin
=-l
F'yn
2X+ I/n
b
2(2A + ,8)(A + #In)
and
ti,l/n
The expected profits of firm i are given by Egin = (X + /3/n)E(xin(si))2 and the
payoff of the first stage for firm i is Pin(45 ..., tn) = Egin - o(ti). We check in the
Appendix that
a 2Pin
<
a t iaP
0
n
at,
(Pin strictlyquasiconcavein ti), and that
a
atiatj
<0
for j *i
866
XAVIER VIVES
(best responses are downwardsloping).Thereforethere is a unique symmetric
equilibriumt*"= t * i= 1,..., n, using a similarargumentto Lemma 1 in Li,
McKelvey,and Page (1985).Let
On(t) =
apin
|=t'
j=1,..n
The unique symmetricsubgameperfect equilibriumof the two-stagegame tn*
solves in t cp(t) = 0 provided the solution is positive and equals zero otherwise.
Some computationsshow that
9
2(X + /3/n) (l + (n - 1)-y) + (n - 1)-yt,B/n
2
On(t
+
+
[2(XA #In (1 (n 1)-y) (n 1)yt/p/n]3
1
c
a2 (1t)2
where
1t/n
2( +#In) - tln
The solution t * will be positiveif a 2> 2,c(X + /3/n). Lettingn tend to infinity
one gets from OpO(t)=0
11t
c
22
X
,2 A+ i t }
Solving for t we obtain
o2_
(J2 + ,B
We have shown thus that t * convergesto
t*=max{O,
as n goes to infinity.
a2_2g
a
2+fi
1c
}
Q.E.D.
REMARK1: In the constantreturnscase it follows from the proof that firms
will purchaseinformation(t* > 0) if n is largeenough(since a2 > 2c1f3/n for n
large enough). Furthermoreit also follows easily from the expressionof 'pn(t)
that t * will be decreasingin n for large n and will eventuallyconvergeto 0.
With many firmsand constantreturns,all firmspurchasea vanishingamountof
information.The averageprecisionin the marketconvergesto zero as n goes to
infinity.
REMARK
2: From the formula for 1/v* it is immediatethat the precision
purchasedis monotone in its cost. If c is zero, firms get perfect information
867
AGGREGATION OF INFORMATION
(t* = 1 or v* = 0); as c increases t* (or 1/v*) decreases monotonically until c is
so high that no informationis purchased.Similarly,the equilibriumprecisionis
monotonicallyincreasingin the priorvarianceof demand(a2). Moreuncertainty
induces the firms to acquirebetter information.The relationshipbetween the
slope of marginalcosts (2X) and 1/v* is not monotonic.For X small and for X
large expectedprofitsat the marketstage are low and consequentlythereis low
expenditureon research.1/v* peaks at intermediatevaluesof X. It is increasing
in X for low values of X and decreasingfor high values.
Welfare
What is the firstbest outcomewith costly acquisitionof information?It is just
the full informationfirst best. In the continuumeconomy we can think of a
center achieving the full informationfirst best by pooling a continuumof iid
signals of zero precisionfor which E(silO)= 0 since (conditionalon 0) the true
value 0 is obtained (almost surely)even if the errorterms of the signals have
infinite variance (this is just the statementof Kolmogorov'sS.L.L.N. for our
continuum economy).In terms of the replicamarketsas n goes to infinity the
optimalexpenditureon informationconvergesto zeroin per capitatermsand the
precisionof the aggregatesignalgoes to infinity.11
In the continuumeconomythe first best per capita expectedtotal surplusis
given by ETS0
=
bEO2/2 according to Table I, where b = 1/(2X + ,B). With
decreasingreturnsto scale,,a competitivemarketalways falls short of this first
best level unless the cost of informationis zero. If the competitive market
expends a positive per capita amounton information,it cannot attain first best
efficiency,which involves zero averageexpenditureon research.If the market
does not buy any informationthen the precisionof signalsis zero and again an
inefficient outcome obtains. The welfare loss is given by 2 (b - a)a2 + c/v*
where a = t*/(2X + ,Bt*), v* and t* are as in Proposition 5. The welfare loss is
always positive except if c = 0.
1"Considerthe n-replicamarketand supposethereis a centerwhichpurchasesan aggregatesignal
with precision l/wn. Given the pooled signal Sn expectedper capita total surplus(gross of
informationcosts) conditionalon Sn is E(601n)x- (/3/2 + X)x2 where x is averageproduction.
Sn
Optimal production is given by x0(sn) = E(6190)/(P + 2X). Expected total surplus as a function of
the precisionof the signall/wn is then
n
~~~~2
ni
C
2=)E(E(Is))
2
)
C
2(
+ 2X)
wn
2(P2X+ wn). The optimalpurchaseof informationprecisionis easilyseen to be
where Tn= a2/(U2
wno
2(
+2X)c
-J?)
which convergesto infinityas n goes to infinity(and Tnoconvergesto 1 as n -- oo). Neverthelessthe
expenditureon informationin per capita terms convergesto zero as n goes to infinity since
(c/nwno) -.-n 0. In the limitper capitaexpenditureon informationis zero and the full informationfirst
best is achieved.We can reinterpretthe centerin the n-replicamarketas collectingn signalswith
precisionl/ln and poolingthemto form Sn-
868
XAVIER VIVES
With constant returns to scale the analysis of market optimality is more
delicate. We have found that in the continuum limit no equilibrium exists but
neverthelessfor any finite n equilibriumdoes exist and involvesa vanishingbut
positive averageexpenditureon information.The questionis: whatis the limit of
per capita total expected surplusas we replicatethe market?From the proof
of Proposition5 and Remark1 it is easily checked that the total precisionof
informationin the marketgoes to infinityas we replicatethe economy(it is of the
orderof magnitudeVH)whilethe averageexpenditureon researchgoes to zero (it
is of the order 1/ Vn). This implies that in the limit first best efficiencyis
obtained!'2We can gain some intuitionfor this result as follows: Fix a large n;
we know then that everyfirmbuys a little bit of information.If firmswereprice
takers almost an efficient outcome would obtain accordingto the results in
Section 4; with constantreturns(and the affineinformationstructure)the market
aggregatesinformationand average expenditureon informationis small. By
increasingn firms eventuallybecomeprice takers;they alwaysbuy some information, and the averageexpenditureon researchgoes to zero. The limit is thus
the first best outcome.Li, McKelvey,and Page (1985) consideralso information
acquisitionwith a constantreturntechnologybut they do not replicatedemand
when the number of firmsincreases.They find that the aggregatepurchaseof
informationmay be positivein the competitivelimit even thoughindividualfirms
purchases are zero. The welfare analysis indicates that the total amount of
researchundertakenby the marketcan be eithertoo much,if the cost of research
is low, or too little, if the cost of researchis high. It is worth remarkingthat,
when examining the convergencepropertiesof equilibriawithout replicating
demand, the "limit"marketis not well defined.Considerthe certaintyCournot
model with constraintmarginalcosts: if demandis not replicatedwhen increasing the numberof firms,individualoutputs are zero in the limit but aggregate
output equals the competitiveoutput. The contrast of our results with those
12With constant returns to scale Lemma 1 gives us the equilibrium strategies given a common
precision t:
an (S, - I)
Xn (S,
+ btL
with
nt
an
a
~~andb=-
A(2+n(-1)t)
1
2,8
Substituting in the expression for per capita total gross surplus and taking expectations, we obtain:
EGSn =
n(2 + n)
y2(1?n)23
nt((n - 1)t + 3) a2
2
2+ ((n-1)t?2)2
Therefore per capita expected total surplus at the market equilibrium t* equals
ETSn=EGSn-2
= EGSn -c
ETSn
n
1 t*
-t*
As n goes to infinity t * converges to zero and nt * to infinity. This implies that ETSn
a2)/2#, which is the first best level.
n
G(2 +
AGGREGATION
OF INFORMATION
869
obtained by Li, McKelvey, and Page (1985) highlights the importance of the
replica assumptions made when studying convergence issues.
With decreasing returns nevertheless the market does attain the "second best"
where expected total surplus is maximized given that firms can base decisions
only on their private information. The intuitive explanation is as follows. From
Proposition 2 we already know that the market is second best optimal at the
second stage, given the expenditures on research. At the first stage there is no
private information and with negligible players (a continuum of them) each firm
has the right (second best) incentives to purchase information. Proposition 6
states the result.
PROPOSITION 6: With endogenous informationacquisition a competitive market
with increasing marginal costs works like a team which chooses expenditures on
information and decentralizedproduction rules to maximize total expected surplus.
PROOF:Let X > 0. Contingent on a certain expenditure on research to get a
precision of information t we know from Proposition 2 that the market works
like a team to maximize expected gross surplus (gross of the cost of information)
and therefore both attain EGS(t) = (aa2 + bMt2)/2,where a = t/(2X + ,Bt) (from
Table I). We show now that the market solves the program MaxEGS(t) - co(t).
The market t* is determined by the symmetric F.O.C. of the problem MaxE7To(ti), which yields:
A 2
(2X.+ ,Bt)2 -
()<
O
with equality if the solution is positive. This is identical to the F.O.C. of the
problem MaxEGS(t) - co(t), which is also sufficient for a maximum to obtain.
Q.E.D.
REMARK: According to Proposition 6 there is no room for a planner to
improve on market performance taking as given decentralized decision making.
This contrasts with Laffont's analysis of rational expectations equilibria under
asymmetric information (Laffont (1985)). In his model, with a partially revealing
rational expectations equilibrium, there is scope for public intervention since
individual private research decisions do not take into account the externality
effect on the informativeness of prices. The externality issue arises since in the
rational expectations model agents do condition on prices.
As a corollary to Proposition 6 we obtain that the welfare loss with respect to
the first best is increasing in the cost of information. This is easily seen since with
increasing marginal costs, the competitive market acts as if it were solving the
team program Maxcj(v, c) = EGS(v) - c/v. Therefore, using the envelope condiv
870
XAVIER VIVES
tion,
dcp(v*, c)
d
dc
_do
dc
(V c).
This equals - 1/v*, and the net expectedtotal surplusof the marketdecreases
with the cost of informationas long as v* < oo. Proposition7 summarizesthe
marketperformancewith respectto the firstbest with decreasingreturns.
PROPOSITION 7: With endogenousand costly informationacquisition a competitive market with decreasing returns always falls short of first best efficiency.
Furthermore, the welfare loss is increasing in the cost of informationas long as the
market expenditure on research is positive.
It may be thought that the inefficiencyarises because we have modelled the
market period as a one-shot affair and firms cannot learn about demand
conditions from prices.We arguebelow that this is not the case.
6. REPEATED COMPETITIONAND INFORMATIONREVELATIONTHROUGH PRICES
Supposenow that we add anothermarketperiodto our gamein the continuum
economy with increasingmarginalcosts. Firmspurchaseinformationat the first
stage, compete in quantitiescontingenton the received signals at the second,
observe the marketprice, and compete again in the last period. We claim that
each firm purchasing precision t* at the first stage, producing x(si) = a(si - u) +
b,i at the second where a = t*/(2X + ft*) (both as in the two-stage game),
inferring9 from the marketprice, and producingx = b9 at the third stage is an
equilibriumpath for the three-stagegame.
To prove our claim supposethat firmshave alreadychosen their precisionof
the information, ti, i E [0,1]. Firm i at stage two will follow the strategy
xi(si) = ai(si
-
u) + bu as in our previous two-stage game since in the continuum
economy the firmcannotaffectthe marketprice throughits quantitychoice.The
market price conditionalon 9 will be then p = 9 - IZ(9) where x(9) is the
average output, x(9) = d(9 - ,u) + b,u with a = faidi = t/(2X + /3t) and t-=
ftidi.13 Therefore p = 0(1 - ,iB) - /3(b - a-),u and provided that 1 - fa- O0, 9
can be obtained by observing the market price: 9 = (p + ,B(b - a-),)/(1 - 3a-).
af3 is always less than 1 except if X= 0. When 9 is revealeda full information
competitive equilibrium obtains at the last stage, x = b9 and p = 2Xb9.
The incentives to purchaseinformationin the three-stagemodel are identical
to those in the two-stagecase since, given the precisionof the informationof the
13To obtain the result use the fact that conditional on 9, ft,s, = t- by analogy to the fact that
given 9,
1
t
n I
tn
converges almost surely to 9 as n goes to infinity since E(s-nIO)=
Var(Sn I) - nO since Var(snIO))< a2/4n.
and it is easily seen that
AGGREGATION OF INFORMATION
871
firms at the second stage, the same equilibriumas in the two-stagecase obtains
and the equilibriumat the third stage does not dependon previousinformation
purchases.We have thus the sameequilibriumbehavioras beforeat the firsttwo
stages followed by a full informationcompetitiveequilibriumsince the market
price is fully revealing.
It is worth noting that at the last stage a fully revealingrationalexpectations
equilibriumprevailsbut neverthelessfirmshave an incentiveto purchaseinformation since this affectsexpectedprofitsat the second stage before the market
price is observed(see Dubey,Geanakoplos,and Shubik(1982)for an elaboration
of this idea in a strategicmarket game context where they show that Nash
equilibriaof the continuumeconomyare fully revealinggenerically)."4
According
to our model in largemarketswherefirmslearn from past pricesthe numberof
firmswho choose to acquirecostly informationneed not be small.In fact in our
symmetric equilibriumall firms purchaseprecision t*. We see thus that the
marketinefficiencyprevailswhenfirmscan learnfromobservedpricessince their
informationcontentbecomesonly availablethe next periodof economicactivity.
7. CONCLUDING REMARKS
We have examinedinformationaggregationin large Cournotmarketswhere
firms receive private signals about the uncertaindemand and argued that the
appropriateapproachis to consider replica markets since that yields a well
defined limit competitivemarket. Bayesian-Cournotequilibriaare very easily
characterizedin the continuummarketand closed form solutions can be obtained.
Our main contentionis that largeCournotmarketsin generaldo not aggregate
informationefficiently.In a competitivemarketwherefirmsreceiveprivatenoisy
signalstherewill be a welfareloss due to the incompleteinformationexceptif the
technology exhibits constantreturns.The marketwill underproduceif the true
state of demandis high and will overproduceif it is low. Our analysisindicates
that the full informationcompetitive(or Walrasian)model need not be a good
approximationof a large Cournotmarketwith incompleteinformationif there
are decreasingreturnsto scaleand if the precisionof the informationof the firms
is not close to perfect.
With decreasingreturnsto scale,if the informationdecisionis endogenousand
costly, then a furtherreasonfor marketinefficiencyis addedand the welfareloss
was shown to be increasingin the cost of information.With constantreturnsto
scale, some subtle existenceissues arise which underscorethe importanceof the
replica assumptionsmade for the welfareanalysisof marketswith endogenous
acquisitionof information.The incentivesto purchaseinformationin a competitive market were also shown to be invariantto adding anothermarketperiod
where firms compete after having observed the previous market price (which
turns out to be fully revealing).
"40n this and relatedissues, see also Blume and Easley (1983), Grossmanand Stiglitz(1980),
Hellwig(1982), and Kihlstromand Postlewaite(1983).
872
XAVIER VIVES
Althougha competitivemarketwill not attainin generalthe (full information)
first best, we have arguedthat the marketallocationis indeedefficientif we insist
on decentralizeddecisionmakinggiven the privateinformationstructureof the
economy.A competitivemarketunderincompleteinformationworkslike a team
where the commonpayoffof the decisionmakersis expectedtotal surplus.
Many extensions and refinementsof the analysis presentedhere could be
made. Our models have to be taken as a first step towardsan understandingof
competitivemarketswith incompleteinformation.Let us mentiona few possible
developments.(a) It wouldbe interestingto examinea class of modelswherethe
S.L.L.N. does not hold, due to the correlationpatternof the signals,for example.
In a relatedvein Rob (1987)has developeda model of privateinformationwith
persistentrandomnessin the limit. (b) Anotherinstancewhere the information
aggregationissue is of relevanceis a situationwherefirmsare negligiblebut they
still have some monopolypower,that is, a monopolisticcompetitionworld.This
is explored in Vives (1987). (c) The inefficiencyof competitive markets in
aggregatinginformationwill have consequencesfor the welfareanalysis of the
long and medium run decisions of firms as capacity investment,technological
choice, R&D, and advertising.Someresultsare reportedin Vives (1986b).
Department of Economics, Universityof Pennsylvania, Philadelphia, PA 19104,
U.S.A.
and
Institut d'AnadlisiEconomica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.
ManuscriptreceivedJanuary 1986; final revision receivedDecember, 1987.
APPENDIX
LEMMA 1: Consider an n-replica market with inverse demand and costs given by p = 9 - fX/n,
C(x,) = Xx2(X > 0) and let the information structure be affine, firm i having informationprecision
ti e [0,1]. Then the Bayesian-Cournotequilibria of the game are in one-to-one correspondencewith the
solution to the team problem with objectivefunction E{ Gn(x; 9)} where
-
Gn (x; 0) =OF
E
+ +X)j2n
J '
_
XI
iI
Ix
and there is a unique Bayesian equilibriumwithfirm i producing according to
X,n
(5 ) = ain (Si-F
+ bnA
where
a
fori=
a
1,..., n.
n |
)-
%n
y
E
2X+#/In
| E bn= 2(2X +
f)(X + /n)
dtI/n
and
% 2(X + /n) - t,f/n
873
AGGREGATION OF INFORMATION
PROOF:We first check,as in Basarand Ho (1974), that the Bayesian-Coumot
equilibriaof the
game are in one-to-onecorrespondence
to person-by-person
optimizationof the team functionGn.
This is clearsince G,(x) can be writtenas 7T,(x)
+ f, ((xj), $,) where
=0
Xf((Xj)jz*J
xj ( nn + A) E
XJ-
2n
*
j*
XkXj.
k:*j
k, jzi
The same outcome obtains by solving max, E(s7TIs,)or max E(G Is,) since f,((x1)1 ,) does not
involve x,. Person-by-person
optimizationis equivalentto globaloptimizationof the team function
accordingto Theorem4 in Radner(1962)sincethe randomtermdoes not affectthe coefficientsof the
quadratictermsand it is easilycheckedthat the teamfunctionis strictlyconcavein x. Furthermore,
with the affineinformationstructure,conditionalexpectationsE(9ls,) are affinein s,. Theorem5 in
Radner (1962) implies that there is a unique Bayesianequilibriumand this equilibriuminvolves
strategiesaffinein the signals.To checkthat the candidatestrategiesform an equilibrium,note that
expectedprofitsof firm i conditionalon receivingsignalsi and assumingfirmj, j $ i, uses strategy
are
x.
i)
E(f,ls,)= x, (E( Ols ) -nEE(X(J)
(n
))
First orderconditions(F.O.C.)yield then
) xX(s,)=E(9Is,)--
2-+
EE(x1(S,)Is,)
for i=1,...,n.
By pluggingin the candidateequilibriumstrategyit is easily checkedthat they satisfy the F.O.C.
(whicharealso sufficientgivenourstructure).
Q.ED.
COMPLEMENTTO THE PROOF OF PROPOSITION5: We check that
2
at2
|
PZn
< ?
At,
and that
( d d2
E7,n=(
n(t)
for j*i,
)<0
+?X)E(X,ln(St))
n.
i1.
=(ba
n+(A)
fl(t))(
+?X)
where
ti 1+ ,Yjn)
using the fact that xin(S,) =a
t)+b,t where ain,bn and Yinare as in Lemma1. To ease
n(s,- since n is fixed.It is easilycheckedthat
notationwe drop all the n subscripts
t.
2)
F(t)=
(ckG ?a,)
2(X + /n)
#In / -
where
-A,t,)2
and A
/3/n~~~~~j
874
XAVIER VIVES
Now,
c
t.
Pi ( t) = Effi - a21
ad
(Pi
)(
+=1-
at,
ti'
2
n
c
2F
III
a
JnJ
1
at, a2 (1-_ti
)2
and
aF
+(1 + Ai) + Aiti
adti=
a2Pi
at2 =
+,lA,)
-Aiti)3'
na2 4Ai(24(1+Ai)+A,ti)
3
2
c
(.p(1+Ai)-A,ti)4
a2 (l-ti)3
and
a2 ,
at,
2c
1
Ai(20(l +Ai)+Ajtj)
a2 (1 -ti)2
\
+1+a)-a)
+(l+Ai)
1)
-
1 - ti
+ aiti)
'
which is negative since
Ai(1 - t,)(24p(1 +A,) + aiti) -
(42(1
+A2
-2_At,2
24p(1 - t,)(1 +Ai)A, +yA(j - ti)ti +A2ti2_-4(1 + Ai)2
which is less than
24p(I -ti)(A,
+ A2iti- 24p(l + 2_i +_2i)
+_i
(2=-20
+ t, - 20) + _ i(24P(1 -ti ) -4+ ) -24
2ti
2
a2p
atiaty
n+
a {a
a2F
atiatj
8)a
\F
O.
a
dtjat'
aAi
aAi at, at.
dzl
a ' > 0 from the definitionof A, and yj, and
a(a dF)
d_,
dt;
(++
ti)(4(i
+-i)-_,t,)
{^t1
-3(| A
t)(4(1
+Ai) +Ait)
8-A.t.8~~~~~~~~~~~~~~~~~~~~~
AGGREGATION OF INFORMATION
875
which is negative since
4+ti<4+(24-3ti)=3(4-ti)
and
4>2,
ti<1.
We conclude thait
(
;
dtj dtj}
Q.E.D.
<o.
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