Bolton Farrell (1990) - Decentralization Duplication and Delay

7/31/2019 Bolton Farrell (1990) - Decentralization Duplication and Delay

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Decentralization, Duplication, and DelayAuthor(s): Patrick Bolton and Joseph FarrellSource: The Journal of Political Economy, Vol. 98, No. 4 (Aug., 1990), pp. 803-826Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/2937769

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Decentralization, Duplication, and Delay

PatrickBoltonEcole Polytechnique

joseph FarrellUniversityof California, Berkeley

We argue that although decentralizationhas advantages n findinglow-cost solutions, these advantagesare accompanied by coordina-tion problems,which lead to delay or duplicationof effort or both.Consequently,decentralizationis desirable when there is little ur-gency or a great deal of private information,but it is strictlyunde-sirablein urgent problemswhen privateinformationis less impor-tant. We also examine the effect of large numbers and find thatcoordination problems disappear in the limit if distributionsarecommon knowledge.

I. IntroductionMost economists believe that decentralized economic systems aremore efficient than centralized ones. But the theoretical basis for thisbelief is not entirely clear. Certain theoretical results-notably thewelfare theorem and the Coase theorem-state that, in idealizedmodels, decentralized outcomes are Pareto efficient. But as Lange,Lerner, and others noted long ago, even if we accept the assumptionsrequired for these results, this alone cannot justify a preference fordecentralization since centralization might be equally efficient.

We thank Philippe Aghion, Kenneth Arrow, Eddie Dekel-Tabak, Mathias Dewatri-pont, Bernard Elbaum, Oliver Hart, David Kreps, Leo Simon, George Stigler, JosephStiglitz, Martin Weitzman, Michael Whinston, and, especially, Paul Milgrom, SuzanneScotchmer, and Jean Tirole for helpful comments. Farrell thanks the National ScienceFoundation (grant IRI-8712238) and the Hoover Institution for financial support.[Journal of Polatcal Economy, 1990, vol. 98, no. 41? 1990 by The University of Chicago. All rights reserved. 0022-3808/90/9804-0002$01.50

803


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804 JOURNAL OF POLITICAL ECONOMYIn response, Hayek (1945) argued that the missing element is anunderstanding of how decentralized and centralized systems dealwith what we now call private information. He suggested that, in

informational terms, an economic system must either put "at the dis-posal of a single central authority all the knowledge which ought to beused but is initially dispersed among many different individuals, or[convey] to the individuals such knowledge as they need in order toenable them to fit their plans in with those of others" (p. 521). Hayekargued that centralization does the former poorly and that a decen-tralized market system does the latter well, notably through prices.As we know, however, the price system does not always work per-fectly. For instance, prices may fail to reflect some "external" costs. AsPigou pointed out, this deficiency may in theory be repaired by cen-tral intervention in the form of corrective taxes. But if the centralauthority does not have the (considerable) information needed tocalculate the right corrective tax, a clumsy intervention may do moreharm than good, and it may be better on balance to put up with theexternality. Whether this is so depends, of course, on just how igno-rant the central authority is, on how harmful the externality is, and onother factors.Another, much less studied, problem that a laissez-faire system maynot solve perfectly is coordination. Quite different from ordinary ex-ternality or monopoly problems, coordination problems involve morethan one relatively desirable (pure-strategy) equilibrium, and otheroutcomes are undesirable. Examples include Schelling's (1960) well-known where-to-meet puzzles, problems of the choice of compatibilitystandards, and the problem faced by two potential entrants into anatural monopoly. In these problems, prices alone typically do notguide actions well. For instance, in the entry problem, the desirableoutcomes involve one firm entering and the other not. Because this isan asymmetric outcome of an initially symmetric problem, anony-mous market prices cannot suffice.As with externalities, simple complete-information models wouldsuggest that central intervention could easily solve these problems.Indeed, in this case, no taxes or coercion would be needed: merely"indicative planning," in which the planner makes suggestions with-out coercive power.' But just as we should not take Pigou's analysis asproving that central intervention is necessarily desirable, so here, if acentral planner lacks important private information (for instance,about the relative merits of the equilibria), intervention may be unde-

' For example, this is true in the coordination problems analyzed by Schelling (1960),Dixit and Shapiro (1986), Farrell (1987), and Farrell and Saloner (1988).


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DECENTRALIZATION 805sirable. In this paper, we study an example of a coordination problemwith private information and show how the desirability of unin-formed central coordination varies according to the importance ofprivate information and the importance of coordination.In our example, two firms contemplate sinking costs in order toenter a natural-monopoly market. We suppose that these costs areprivate information, so that neither firm (nor anyone else) knowswhich has the lowest costs. The best outcome is that the lower-costfirm enters and the other stays out. "Decentralization" in our exampleis an incomplete-information game of timing between the firms: Inany period, if neither firm has already sunk the costs of entry, eachdecides whether to sink costs ("enter") or to wait another period.Waiting is sensible if the firm fears that the other is likely to enter.In a symmetric equilibrium of this game, each firm is uncertainabout when the other will enter. Lower-cost firms are less worriedabout the possibility that their rival will enter and therefore choose toenter sooner than higher-cost firms do; consequently, if the firmshave (sufficiently) different costs, then the lower-cost firm enters andpreempts its higher-cost rival. Thus the laissez-faire system effectively"uses"private information, even in this model without prices. But thisdecentralized coordination is far from perfect. If the firms have equalcosts, then they enter simultaneously: there is inefficient duplication.Ifboth have high costs, both wait: there is inefficient delay.In our example, our model of "central intervention" consists ofnominating one of the two firms to enter; in the spirit of Hayek, wesuppose that (for reasons we do not model) a central planner cannotor will not collect and use the private information in the way thatmechanism design theory might suggest. In other words, we model aclumsy central solution to the coordination problem. We compare theexpected social cost (relative to the first-best) of the fact that half thetime, the uninformed central planner will nominate the higher-costfirm to enter, with the cost of duplication and delay incurred underdecentralization.Not surprisingly, we find that the less important the private infor-mation that the planner lacks and the more essential coordination is,the more attractive the central planning solution is. Perhaps moreinteresting is the effect of urgency. The decentralized solution, sinceit requires time to work, performs poorly if time is pressing. Centrali-zation-in our model-does not involve delay and consequently is agood system for "emergencies," when delay is very costly. This predic-tion of our model is consistent with the fact that almost all organiza-tions rely on central direction in emergencies. In particular, in war-fare, laissez-faire decentralization is little favored: armies work by


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8o6 JOURNAL OF POLITICAL ECONOMYcommand. Studying economic organization in World War II, Mil-ward (1977, p. 99) found that

the differences between the political societies involved in theSecond World War were very wide. Once commitment to awar involving a high level of economic effort was accepted,however, the economic problems to be solved were oftenvery similar. A comparative study of the machinery by whichthese economic decisions were taken and enforced indicatesthat common decisions impose to some extent common ad-ministrative solutions.... Everywhere heprice mechanism ameto beregardedas a methodof allocating resourceswhichwastoo slowand too risky.[Italics added]Similarly, Scitovsky, Shaw, and Tarshis, in their study of economicmobilization for war (1951, p. 139), found that the market system"takes time-sometimes a considerable length of time-in effectingadjustments to changes in supply and demand conditions."In this paper we use an example to address two questions that wethink receive too little attention from economic theorists. The firstconcerns adjustment costs: How do different mechanisms performwhile approaching equilibrium?Our example illustrates some of theinefficiencies that may arise as a decentralized system gropes for anefficient static "equilibrium" in which high-cost firms are "out" whilelow-cost firms are "in." In particular, the time required for decen-tralized screening may be costly. The second question concerns coordi-nation: If agents simply do not know what others are expecting themto do, then coordination problems arise, and these are not well cap-tured by standard notions of externalities and incentives (which de-scribe why agents may knowingly act in a way inconsistent withefficiency). As our example suggests, central direction may resolvesuch coordination problems, but probably imperfectly.

II. A Model of Decentralized Coordinationin EntryWe consider the following simple coordination problem. Two firms,A and B, are contemplating entering a natural-monopoly industry.Entry involves a sunk cost, S, which is borne by the entrant. A singleentrant will capture a fraction A of the gross social benefits of entry,V, which we normalize to be one, so that a sole entrant's net payoff isX - S. We suppose that each firm's value of S is known only to thefirm itself but is independently drawn from a known distribution F( ).If both firms enter, each bears its cost S, and each can appropriate a


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DECENTRALIZATION 807fraction Rx f the social gross benefit V = 1.2 (For instance, in the caseof entry into a natural-monopoly industry, with Bertrand competitionshould both firms enter, we have 0 < X c 1 and ii = 0.) We supposethat pu< S < X for all S in the support of F( ), so that this is acoordination problem.3 Each firm wants to enter if and only if theother does not enter. There are thus two pure-strategy equilibria;other outcomes are inefficient. The pure-strategy equilibria are typi-cally not equally desirable, however: it is better to have the lower-costfirm enter.We take "decentralization" here to mean that, at any date, each firmchooses to enter or not, with no explicit consultation with the other. Ifneither enters, then once each sees that the other has not entered, thegame begins again (although with different beliefs). It is natural tosuppose that there is a lag between a firm's irreversible commitmentto sink the entry costs S and the unambiguous observation of thatcommitment by the other firm. We model this by supposing that thereare infinitely many "periods" 1, 2, . . . and that in each period eachfirm observes what happened in the previous period and then chooseseither to enter or to wait another period. Once entry by at least onefirm has occurred, production takes place and the game ends.Benefits are discounted at a constant rate 8 per period, both privatelyand socially.4In this game, there are asymmetric subgame-perfect equilibria that"solve" this coordination problem: for example, firm 1 always entersimmediately and firm 2 never enters. Much work on decentralizationhas focused on the constraints imposed by incentive compatibility,and if we were to take that approach, we would find that decentraliza-tion was consistent with these asymmetric equilibria. But it is far fromclear how the firms would "find" one of those equilibria: we cannotassume that they do so without some process of explicit coordination.Moreover, those equilibria do not capture the informational benefitsof decentralization since the private information on costs does notaffect which firm enters. In analyzing decentralization, therefore, wefocus on symmetric (Bayesian) equilibrium.5

2 If competition ex post makes for a more efficient exploitation of the social opportu-nity modeled, then the grosssocial gain from the entry of both firms may exceed V = 1.But unless it exceeds I + S, duplication is still socially undesirable.3 This is assumed for ease of exposition only; in an earlier draft, we considered themore general case in which some values of S may lie above X or below p.4 It would be possible in principle to model these delays within a continuous-timemodel; however, we suspect that it would be technically more complex and would yieldlittle additional insight.5 Dixit and Shapiro (1986), Farrell (1987), and Farrell and Saloner (1988) arguedinformally that asymmetric pure-strategy equilibria are unconvincing and inappropri-ate for the study of decentralized coordination mechanisms. See Crawford and Haller

(1990) for a more formal argument.


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8o8 JOURNAL OF POLITICAL ECONOMYTABLE 1

FIRMBFIRMA Enter WaitEnter [t-SA, I -SB - SA,OWait 0, X - SB BV(SA), 6V(SB)

As long as no entry has occurred, the payoff matrix in any periodcan be written as shown in table 1, where v(S) is the continuation valueof the game to a player of type S. We now show (i) that, in equilib-rium, firms with lower values of S enter sooner (in a sense that wemake precise) and (ii) that the process of entry has a declining hazardrate. Proposition 1, which fulfills claim i, is a statement about admis-sible strategies for a single firm; our other results assume a symmetricBayesian equilibrium.

PROPOSITION 1. Low-cost types enter sooner than high-cost typesdo, in the following sense. Fix (say) firm A's beliefs about the (normal-form) strategy of firm B, and consider two possible values of SA, saySX and S25. Suppose that S' < S2, that S' puts positive probability onentering in period tI, and that S2 puts positive probability on enteringin period t2. Then t1 s t2.Proof. A normal-form strategy for firm A is a mixture of the ele-mentary strategies s(t): enter in period t if B has not entered beforethat. If firm A puts positive probability on s(t), then its expectedpayoff from s(t) must be at least as great as that from s(t') for anyother t'. Now suppose that A believes that there is a probability o(t)that B will not have entered before time t and a "hazard-rate" proba-bility h(t) that B will enter precisely at time t (if not preempted). Writea(t) bta(t). Then A's expected payoff from the strategy s(t) is just

a(t)[X - S - h(t)(X - R)].If the type S' puts positive weight on strategy s(t,), then we have

a(t)[X -S - h(t1)(X - ?) a(t2)[AXS- h(t2)( -and

a(t2)[X -S -h(t2)( - )] a(t1)[X - h(t1)(X -Adding, we get

[a(t1) - a(t2)](SA- SA) - 0Now a(t + 1)Ia(t) = 8[1 h(t)], so that a(t) is strictly decreasing in t.Since SI < S2 by hypothesis, this gives the result. Q.E.D.


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DECENTRALIZATION 809COROLLARY A. In a symmetric Bayesian equilibrium, ifjust one firmenters, it is the lower-cost firm.COROLLARY B. In a symmetric Bayesian equilibrium, if the firms'

costs in fact differ, the lower-cost firm surely enters; the higher-costfirm may or may not enter.Proposition 1 and its corollaries show how decentralization sortspotential entrants: if duplication is avoided, low-cost rather thanhigh-cost firms enter. But this sorting is achieved only with delay andonly through the threat of duplication, which makes high-cost firmshang back and allow low-cost firms to go first. And, of course, thisfear is sometimes realized; the likelihood of that is determined by thefirms' strategies. We turn next to a description of those strategies.From now on, we assume a symmetric Bayesian equilibrium.TheFundamental Difference EquationBy proposition 1, there exist cutoffs SI, S2,.. . such that, for values ofS between St- 1 and St, a firm of type S enters in period t (provided thatits rival has not previously entered). Since the firm of type St is indif-ferent between entering in period t and waiting for period t + 1, wehaveX - St - h(t)(X- ii) = 8[1 - h(t)][X St - h(t + 1)(X- it)]. (1)

Equation (1) is a difference equation relating St, St-,, and St,,. Itdescribes the dynamic behavior of the decentralized equilibrium. Ournext proposition summarizes this behavior.PROPOSITION 2. The hazard rate h(t) is nonincreasing. As t --> o, thecutoff type, St,converges to the upper limit Smax of the support of F(.).

The fraction F(St) of all types that have entered by date t converges toone but is bounded above by 1 - [1 - h(l)]t.Proof. First, note that each side of (1) must be nonnegative since itrepresents the equilibrium expected payoff to a firm of type St, andthe firm can always guarantee zero payoff by staying out. This non-negativity implies (by [1]) that the hazard rate is nonincreasing. Theupper bound on the fraction of firms that have entered by a givendate t follows immediately.If St's equilibrium payoff is zero, then (1) implies that the hazardrate h(T) is constant from date t onward and that ST St for all T ? t.Since St < X, (1) implies that the (constant) hazard rate h(T) mustbe strictly positive. This implies that the fraction F(S,) of all typesthat have entered by date T converges to one as T -> o, and lim7,OC - CmaxST = SIf the equilibrium payoff expressed by (1) is strictly positive, weconsider two subcases. First, if h(t) converges to zero, then every type

of firm can ensure a strictly positive expected payoff by waiting long


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81o JOURNAL OF POLITICAL ECONOMYenough before entering; hence every type eventually enters (if notpreempted) and_S, converges to Smax. Second, if h(t) converges to astrictly positive h > 0, then (as with a positive constant hazard rateabove) lim,,O S = Smax.Q.E.D.Intuitively, if there is a firm willing to enter, then entry will even-tually occur, but it can take an arbitrarily long time. So decentraliza-tion achieves some sorting, but it is typically imperfect and takes time.To say more, we must simplify the model still further; we shall con-sider the case in which the distribution F(.) has a two-point support.III. Two TypesWe consider the simple case in which there are just two cost types,"high" (SH) and "low"(SL), where pu< SL < SH < X.Let the probabilityof SL be q. The expected social payoff in the first-best is then simply

W* = 1 - (1 - q)2SH - q(2 - q)SL.We now consider symmetric Bayesian equilibrium of the laissez-faireentry game.

By proposition 1, there is a date T such that low-cost types enter ator before T and high-cost types enter at or after T. For some parame-ter values there will be a mixture of the two types entering at T, andthe sorting process will be incomplete; that is, sometimes a high-costfirm will enter (at T) although its rival is low-cost. For other parametervalues, there will be complete separation; that is, all low-cost typesenter at or before T and all high-cost types enter strictly after T. Insuch a fully separating equilibrium, decentralization always finds thelow-cost entrant in the sense that, if the two firms differ in costs, thenthe low-cost firm is the unique entrant. Thus decentralization isshown most favorably in such an equilibrium. It is simplest to focus onthe case in which T = 1, so that when there is a low-cost entrant thereis no delay.A Simple Equilibriumwith Two TypesFor a range of parameter values and for a suitable value of p, it is aperfect Bayesian equilibrium for both firms to use the followingBayesian strategy:

If low-cost, then enter immediately. If high-cost, then do notenter in period 1; in each period t ' 2, if there has been noentry before t, enter with probability p.It is easy to show that this "simple equilibrium" exists if and only if

u < SL + 85(1 - q)(SH - SL) < A - q(X - A) s SH s A. (2)


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DECENTRALIZATION 8i iIn such an equilibrium, the outcome of decentralization is as follows.If both firms are low-cost (which happens with probability q2), thenthere is immediate duplication; welfare is 1 - 2SL. If one is low-costand the other is high-cost (which happens with probability 2q[l - q]),then there is immediate low-cost entry and no duplication; welfare is1 - SL. If both firms are high-cost (probability [1 - q]2), then the firstperiod sees no entry and the two firms continue under complete in-formation, as we now analyze.Entry Gamewith Complete nformationSince the symmetric equilibrium involves mixed strategies, each firmmust be indifferent between entering and waiting:

v = p(p - S) + (1 - p)(p. - S) = (1 - p)8v,where p is the probability of entry (for a given firm) in any givenperiod, and v is the continuation value of the game. Since (1 - p)b< 1, v = 0 and

p A S (3)Notice that we require ii < S < X:mathematically so that 0 < p < 1and economically so that neither enter nor wait is a dominant strat-egy.We now calculate the expected social surplus. The probability qtthat nothing happens before period t and that then just one firmenters is

qt = (1 - p)2(t-1) X 2p(1 -p)while the probability rt that both do so is

rt = (1 - p)2(t-1) x p2.Consequently, expected social surplus,

00VVG_

-5t[qt(I - S) + rt(I - 2S)t= 1

is given by_ p(2 - p) __ 2p_ _ (4)

1-8(1 p)2 1 - 8(1 - p)2p(2 p) 1- p21-5(1- p)2]( 5)- 1-5(1-p)2

(5)


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812 JOURNAL OF POLITICAL ECONOMYIn equation (5), we see the losses (relative to the first-best welfare level1 - S) due to delay (the second term) and due to duplication (thethird term). As p -> 1, the losses due to delay converge to zero, as theydo when 8-> 1. But large values of p make the losses due to duplica-tion worse, and large values of 8 do not help there. We can interpret8 -> 1 as representing either a less urgent problem or else shorterperiods (less reaction lag).6We can ask whether the equilibrium value of p, given by equation(3), is above or below the (second-best) optimal level. That is, would adecentralized policy (constrained to symmetric equilibrium) do betterby inducing earlier or later entry on average? Such changes could beaccomplished, for example, by changing Xor pu r S. Simple calcula-tions show that in general the answer is ambiguous. Firms bear all thecosts of duplication and only some of the costs of delay (X < 1); thismight make one think that they would be too cautious. But at thesame time there is an inefficient preemption motive for each firm tomove too fast: it does not take proper account of the reduction in itsrival's expected payoff when it enters. Formally, we have

[1 - 8(1 - p)2]WVG p(2 - p - 2S).Hence, the sign of aWGldp is that of(1 - p - S)[1 - 8(1 _ p)2] - 5p(1 - p)(2 - p - 2S).

In particular, for small 8, aWGlap has the sign ofI - p - S = (1 - +pu)S -

so when a problem is very urgent the decentralized solution may betoo hasty or too cautious, according to whether S $ pu/(1 - X + Du).For large 8, aWGhlp < 0 always: that is, for nonurgent problems, thepreemptive effect dominates and firms are always too hasty. Turningto the effects of different sizes of S, we find that for small S (close to Du)there is excessive haste, while for large S (close to X)there is excessivecaution.The probability of (eventual) coordination (i.e., just one firm en-ters) is 2p(1 - p) 2 -2p 2(S-pA)

2p(1 -p) + p2 2 -p X + S -2'6 Indeed, if we analyzed the problem in continuous time following Simon (1987), wewould find instant entry but with a positive probability of duplication. But we believethat such a continuous-time model would be inappropriate since it would assume thatplayers can react instantly to one another's choices. In reality, there are typically obser-vation and decision lags; we believe that these are better analyzed in a model such as

ours with discrete periods, although in principle one could build observation anddecision lags into a continuous-time model.


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DECENTRALIZATION 813Mean delay, y

Mean duplication,x x=1FIG. 1

which is increasing in S, decreasing in X, and decreasing in pu,as onemight expect: eventual coordination is more likely when firms aremore cautious about sinking costs. Such parameters, however, simul-taneously exacerbate the delay in reaching an outcome. There is atrade-offbetweenavoiding duplicationand avoiding delay.Indeed, we cansee that trade-off quite clearly by expressing the probability of dupli-cation and the mean delay in terms of the single variable p: the proba-bility of duplication is just

2 p,while the mean delay is

00y = E (qt + rt)t- 1 = 71__ p(2 -p)(7

Eliminating p from (6) and (7), we find that(x - 1)2 (8)4x

Policies such as subsidies for entry, which operate through changingp, move us along the curve (8), which is shown in figure 1.This completes the analysis of the complete-information continua-tion game. We now return to the analysis of the simple Bayesianperfect equilibrium in the two-type case. Once the first period has


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814 JOURNAL OF POLITICAL ECONOMYpassed without entry, so that it is common knowledge that both firmshave S = SH, welfare is given by (4), or

_G p(2 -p) 2p SH,1- 8(1 -p)2 1 -85(1 - p)2and p is given by (3) with S = SH. Consequently, the ex ante expectedsocial payoff from decentralization is

WD = q2( - 2SL) + 2q(1 - q)(1 - SL)+ (1- q)25[ 1 - 8(1- p)2 1 - 8( - p)2 SHIwhere p = ( -SH)I(X - j). Equivalently we can write the loss rela-tive to the first-best, W*, as

W* - WD = q2SL + (1 - q) 1 - - )2 SHI+ (1- q)2[1 1 8p(2 - p)_ (1- SH)- 5(1 - )2 HJHere we see the losses due to duplication and to delay. We nextcompare these with the losses from a natural model of a highly imper-fect central planning system.

CentralPlanning withIncomplete nformationWhile decentralized equilibrium in our model often selects efficiententrants, it involves duplication and delay. One of our central pointsin this paper is that, for some parameter values, even a completelyignorant centralplanner outperformsdecentralizationbecause speed andreliability of coordination outweigh the need to find the lower-costentrant. In other cases, of course, the comparison is reversed, forunder decentralization the lower-cost firm always enters (althoughthe higher-cost firm may do so as well!). Evidently, the (gross) advan-tage of decentralization is related to the difference E[S] - E[min(SA,SB)],which is small when the distribution of S is concentrated or whenthe two firms' costs, SA and SB, are highly correlated.Suppose then that a central planner does not use private informa-tion at all, but merely chooses an entrant at random. This will achieveimmediate entry without duplication, but there will be no tendency toselect a low-cost entrant: S will on average take its expected value.In the two-type case, expected welfare under a random planner issimply

WR = 1 - [qSL + ( - q)SH].


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DECENTRALIZATION 815Equivalently, we can write the loss relative to the first-best from ran-dom choice as

W* - WR = q(1 - q)(SH - SL). (9)Comparing Centralizedand DecentralizedOutcomesComparing these expressions for expected welfare is complex in gen-eral. Since we are considering only an example in any case, we shallmake the comparison only for the polar cases 8 = 0 and 8 = 1.For very urgent problems, when 8 = 0, decentralization yields anexpected payoff of

2q(1 - q)(1 - SL) + q 2(1 - 2SL) (10)in the simple equilibrium, which exists when pR SL S qp + (1 -q)A S SH S X.7 We can rephrase (10) to show how decentralization failsto achieve the first-best: the difference between the first-best expectedpayoff and (10) is

W* - WD = q2SL + (1 - q) 2(1 - SH), (11)where the first term is the loss due to duplication and the second termis the loss due to delay. Comparing (1 1) with (9), we find the followingproposition.

PROPOSITION 3. In the two-type case, when there is great urgency(8 0) and conditions (2) are satisfied, decentralization is superior torandom choice if and only if

q(1 - q) (SH - SL) > q2SL + (1 - q)2(1 - SH). (12)Evidently, (12) is less likely to hold when SH and SL are close, that

is, when the private information that decentralization may be able touse is relatively unimportant. Likewise, it is more likely to hold when1 - SH is large (so that delay is very costly) and when SL is large (dupli-cation is costly).Turning to the other polar case, 8 = 1, we find that decentraliza-tion gives an expected social payoff ofWD = 2q(l - q)(1 - SL) + q2(1 - 2SL) + (1 - q)2 P(2 P 2PS)1I (1I p)2As before, it is useful to write this in terms of deviation from the first-best, that is,

W* - WD = q SL + (1 - q)2 SH (- - SH). (13)X\ + SH - 2~Comparing (13) with (9), we find the following proposition.

7 Notice that, for any choice of SL, SH, and q, these conditions hold for suitable valuesof Xand ,u. That is, we can choose the latter arbitrarily and still find genuine examples ifwe are prepared to choose X and pL ccordingly.


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8i6 JOURNAL OF POLITICAL ECONOMYPROPOSITION 4. When there is little urgency (8 1), decentraliza-tion is better than random choice if and only if

q(l - q)(SH - SL) > q SL + (I 'X - S -2 SH) (14)X +SH- 2pLSince SH > pL,comparison of (12) and (14) shows that decentraliza-tion is more likely to be the better of these two systems when there islittle urgency. We thus find some confirmation of the idea that decen-tralization is less desirable when a problem is urgent, as in emergen-cies.

Speedand Delay in BureaucracyIt may seem surprising that our model depicts central bureaucracy asan insensitive but rapid decision maker. Most people would probablydescribe bureaucratic central decisions as slow, and often they are;but the fastest decisions are also made by central authorities.While a central authority surely can make a rapid and randomdecision, it may not always do so. Instead, a planner may try to gatherinformation. One natural way to do so is through an advocacy processof some kind, which we might model as a war of attrition betweenpotential entrants. This conforms to some descriptions of the Japa-nese Ministry of International Trade and Industry, which tries tocoordinate certain entry and major investment decisions, in an envi-ronment in which each of several firms would like to be selected. Asimilar kind of voluntary centralization in coordination was studied byFarrell and Saloner (1988), in the context of the choice of compatibil-ity standards. They found that a centralized committee system is morereliable at coordinating than the "market" system, but is indeedslower. However, in evaluating centralization, one should rememberthat there is a choice of how much information to try to use; it wouldappear that rapid random choice is always an option.8IV. Large NumbersIn this section we show that if we replicate our problem, so that thereare many firms drawn independently from the same cost distribution(which is common knowledge), then coordination is no problem, al-

8 A rapid and random choice might well have maximized social welfare in the choiceof standards for AM stereo broadcasting (see Besen and Johnson 1986). Instead, theadministrative process in the Federal Communications Commission became boggeddown in an adversarial proceeding, and eventually the commission refused to choosestandards. As this example suggests, although rapid choice is an option, it may not betaken even when it should be.


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DECENTRALIZATION 817though of course incentive problems may remain. This suggests thatdecentralization might be most useful in large markets and largeeconomies, not because central planners' information processing abil-ity is limited or because large numbers make collusion less likely, butbecause coordination problems under decentralization are mitigatedas the economy becomes large.We can also interpret this result in terms of prices. Suppose thatcosts are independently drawn from a known distribution. Using onlycommon knowledge, for any given fraction ox,we can find an outputprice p(cx)such that, when all agents who can make a profit at pricep(cx)enter and none who would make losses enter, then the supply is axin expectedvalue. For small numbers, the realized supply will randomlydiffer from its expected value ox,and, as Weitzman (1974) showed, itmay therefore be better to use another means of coordination, such ascentralized quantity setting. With large numbers, however, this prob-lem disappears, and there is an almost deterministic "supply curve."This solves the coordination problem and also solves the "prices ver-sus quantities" problem.In a market setting with large numbers, no centralized quantitysetting (as by a Soviet-style planner) or centralizedprice setting (as by aWalrasian auctioneer) is needed to solve the coordination problem:symmetric equilibrium determines a cutoff cost level such that pre-cisely those with lower costs enter, and the equilibrium condition isthat the (confidently) expected endogenous price is just the one tojustify exactly those entering. Of course, that outcome need not beefficient in general, but any problems are standard incentive or exter-nality problems, not coordination problems.

In a sense, this justifies the textbook treatment of entry into a com-petitive market: the right number of firms enter so that the profits forentrants are driven to zero, and no costly delay is imposed by theprocess. But we stress that, here, this is a result, requiring some restric-tive assumptions.Formally, consider n potential entrants, with sunk costs of entry Sdrawn independently from the same distribution F(-). Suppose thatthe grossbenefits accruing to an entrant if a fraction fof all the firmsenter are b(f), where b is continuous and decreasing inf, with b(O) -Smax and b(1) ? Smin. (In the game above, we had n = 2, b(1) = [u,andb('/2) = X.)As above, the potential entrants play a simple timing game:in each period, each chooses to enter or to wait. Because b(F(-)) isdecreasing, there exists a unique cutoff level S such that b(F(S)) = S.and we have the following proposition.

PROPOSITION . In the limit, as n -* oo, all firms with S < S enter inthe first period, and there is no further entry.Proof. In any symmetric perfect Bayesian equilibrium of the game


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818 JOURNAL OF POLITICAL ECONOMYwith n firms, there exists a cutoff level Sn such that all firms withS K SI enter in the first period, and others wait. Now for i = 1, . .. , n,let xi be a random variable defined by

if firm i enters in period 1I if not,and let en be the random variable E7 I x,. The expected number offirst-period entrants is simply E(en) = nF(Sn). The x-are independentand identically distributed. By the law of large numbers, therefore,for any E > 0,

lim Pr -F(S) = O.n-ag [ fland hence by the continuity of b(Q),or any E' > 0,

lim Pr[ b(m) - b(F(S E)) ' j = 0.To see that this implies Sn __ S, observe that otherwise, for some S =#S, we can take a subsequence of n's such that Sn __ S. But this impliesthat, with E' = 1/21S - S1,say, and for large enough n in the subse-quence, b(enIn) s almost surely very close to b(F(S)). But this cannot bean equilibrium since, for S =#S, S =#b(F(S)). Q.E.D.Thus large numbers may make decentralization work much better.Of course, large numbers may also affect the costs of centralization.In particular, Milgrom and Weber (1983)9 consider a planner whodoes not know the distribution of costs across firms (formally, firms'costs need not be independently distributed) but can samplecosts atdifferent firms. They show that when there are many firms, the plan-ner can almost completely learn about the distribution by sampling avery small proportion of the firms. This suggests that, in a largeeconomy, a central planner may not find it very costly to gather infor-mation: he can learn a great deal at a cost that is very small in terms ofthe economy.'0V. Related LiteratureMany readers will find our ideas reminiscent of Weitzman's (1974)seminal paper. For instance, he wrote the following:

9 Green and Laffont (1979) also consider the effects of statistical sampling on thecosts of information acquisition in dominant-strategy mechanisms in large economies.10 Of course, this works not only with costly direct sampling but also with providinginformational rents for revelation of private information: the individual rationalityconstraints can be ignored for a small sample since society can easily afford to subsidizea few randomly chosen "benchmark" firms to see what is feasible and to control theremainder.


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DECENTRALIZATION 819Suppose that fulfillment of an important emergency rescueoperation demands a certain number of airplane flights. Itwould be inefficient to just order airline companies orbranches of the military to supply a certain number of theneeded aircraft because marginal (opportunity) costs wouldalmost certainly vary all over the place. Nevertheless, such anapproach would undoubtedly be preferable to the efficientprocedure of naming a price for plane services. Under profitmaximization overall output would be uncertain, with toofew planes spelling disaster and too many being superfluous.[P. 478, n. 1]

Weitzman emphasized the static uncertainty about total output thatstems from naming prices when supply curves are unknown. Ourwork differs in two ways. First, it is intrinsically dynamic: we areconcerned not only with getting the wrong supply but with getting thesupply too late." Second, and more fundamental, Weitzman ad-dressed a problem of optimalcentralization:Should a central plannerset prices or set quantities? He (consciously) did not ask whethercentralization is desirable. In a decentralizedmarket system, despite themisleading analogies of Walrasian auctioneers, prices are not cen-trally "named": they emerge over time from asymmetrically informedfirms' initially uncoordinated entry and supply decisions. Thus whenairplanes are needed, for instance, decentralization does not mean an-nouncing a price that the planner hopes or guesses will elicit thecorrect supply any more than it means announcing who should sup-ply which planes; it means waiting to see who will decide (not knowingthe prices that will prevail) to bring planes to market, and the priceand the supply are joint consequences of those decisions.Recent work on decentralized coordination of the kind studiedhere includes Dixit and Shapiro (1986), Farrell (1987), and Farrelland Saloner (1988). These papers studied how decentralized coordi-nation through unilateral decisions is possible but imperfect; Farrelland Saloner compared that with coordination through standardiza-tion committees, a form of voluntary centralization. All those papers,however, studied complete-information models and so could not ana-lyze the "sorting" consequences of decentralization.Crawford and Haller (1990) formally justify the use of symmetricequilibrium analysis in studying coordination. They focus on gamesin which players' interests completely coincide (so that cheap talk, ifallowed, would resolve their problems), but their arguments could beextended to our case." The example above is not the only one in which it seems that Weitzman had in

mind some notion of urgency, but his analysis did not reflect this.


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820 JOURNAL OF POLITICAL ECONOMYBliss and Nalebuff (1984) analyzed decentralized equilibrium withincomplete information (but in a war-of-attrition rather than a grab-the-dollar framework). They showed that equilibrium involves delay

(though, in their continuous-time framework, no duplication). 12 Theydid not, however, draw out the second-best welfare implications oftheir equilibrium analysis.Our work is inspired by the fundamental questions raised in theearly "market socialism" debate. Lange, Lerner, Mises, and Hayek,among others, noted that welfare theorems under complete informa-tion are at most a first step toward welfare comparisons of centraliza-tion and decentralization. Their consensus was that the matter shouldturn on which system best dealt with private information and theproblems that it causes; they debated whether decentralization andthe price system or a central planning board (perhaps using pricelikealgorithms) would do better. Our work shows, in a special model, howdecentralization can use private information even without a pricesystem; but it also warns that market screening involves efficiencycosts. 13VI. ConclusionWe close by trying to put our very simple analysis into a broaderperspective. It seems that laissez-faire systems often efficiently usedispersed private information, but that, in addition to the widelystudied incentive problems (monopoly, externalities) that such sys-tems may encounter, they also may perform imperfectly on coordina-tion problems. On the other hand, it seems that centrally plannedsystems may be better at making rapid and arbitrary choices but dopoorly at gathering and using dispersed information.We have modeled this contrast very starkly in the present paper. Inreality, both centralized and decentralized systems work in more sub-tle, and perhaps more efficient, ways than we have portrayed. Weassumed that the decentralized solution involves no communicationor other private coordination between the firms. In fact, private coor-dination may take place through communication (cheap talk),14through binding negotiations between potential entrants, 5 throughjoint ventures, or through competition for investment capital or other

12 A continuous-time version of our game (analyzed according to the methods ofSimon [1987]) would involve duplication but no delay. We do not stress this point,however, because we think that reactions are not instantaneous, and this is properlyreflected in periods of positive length.13 For a recent related argument, see Stiglitz (1989).14 See Farrell (1987) for an analysis of this and of its limits.1 Such negotiations would have to be very circumspectly conducted if they are not to

violate antitrust laws, however.


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DECENTRALIZATION 82 1scarce factors of production. Similarly, we assumed that a centralizedsystem simply ignores private information. An opposite view is re-flected in the theory of mechanism design, which assumes that a cen-tral authority can gather and use all information, subject only torespecting incentive constraints. In particular, the problem that weanalyzed may be very simply solved by such a planner, simply byauctioning a "franchise."We think that, realistically, these and other mechanisms of laissez-faire coordination are likely to work only imperfectly in a world ofincomplete information. For instance, private negotiations are subjectto the usual inefficiencies of bargaining between privately informedagents. Likewise, if a central planner tries to collect private informa-tion, delays ensue. Collating information is time consuming, and themore information must be marshaled in one decision maker's mind,the slower the process. Because of information processing limits, in-formation must be condensed before being compared. F'Moreover, acentral authority may be only imperfectly able to commit itself to amechanism in advance, and private agents may not be fully com-mitted to the mechanism. For instance, notably in defense procure-ment, bidders sometimes claim to have low costs, take on the contract,and later insist on renegotiating because costs have proved higherthan "foreseen." Even the apparently simple first-best "franchise" so-lution to our entry problem may not work if these commitments arenot available.'7 Because of these difficulties, we believe that the simplestory we have described retains an element of truth even when theseand similar refinements are allowed for.Broad confirmation of these ideas seems to be provided by evidenceon society's reactions to emergencies and to wars. Although manyWestern societies laud the laissez-faire system in peacetime, in anemergency they change their tune. Of course, there are many possiblereasons for this, and our model touches on only one. But it is the oneidentified by Milward and by Scitovsky et al. as the prime defect of amarket system's response to large new opportunities or problems. It isalso consistent with some organizational choices made when speed isimportant but there is no "emergency."'8

16 Geanakoplos and Milgrom (1984) have modeled optimal organizational form tak-ing into account this constraint.17 In the Appendix, we analyze how the central authority might perform in ourproblem if unable to make certain commitments.18 See, e.g., Chandler's (1977) discussion of how firms became more centralized asspeed of reaction became more important. Also, H. Karatsu, of the Nippon Telegraphand Telephone Company, recently explained that the company "used a free biddingsystem until the U.S. occupation ordered it to do otherwise. That was because Japanhad to develop a good telecommunications network system as quickly as possible" (New

YorkTimes,July 21, 1987).


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822 JOURNAL OF POLITICAL ECONOMYWe mention just one further important limitation of our analysis.We assumed that there is a clear social "opportunity" and that theonly question is which of two contenders should exploit it. An alterna-

tive possibility is that, rather than (accurate) private information here,on their own costs), private agents may have differing views onwhether this new market is valuable at all. In that case, it may not beclear how the information should be used (aggregated) even if it wereall public. Under decentralization, entry will occur if someagent is suf-ficiently optimistic (allowing, presumably, for a winner's-curse effect):an extreme-value statistic of the sample of views. When is this a goodrule, and compared to what alternatives? Sah and Stiglitz (1985) havebegun to address this question, but much remains to be done.In this paper, we have tried to formalize some ideas of the "Aus-trian school" on how decentralization brings private informationto influence market outcomes. Although such discussion is usuallycouched in terms of market prices, we believe that prices are not theessential element. As in our model, private information often affectsdecentralized outcomes even without prices. But the link is typicallyimperfect, and sometimes (especially, we argued, in urgent problems)the imperfections outweigh the advantages of decentralization.Appendix

Central Direction versus Decentralization andMechanism DesignAs Langeand Lerner pointed out long ago, from the pointof viewof mecha-nism design theory, decentralizationshould never do better than a sophis-ticated planner: for the latter could always replicate the marketinstitutionand mightdo strictlybetter.In fact,in our model she can implementthe first-best by auctioninga franchise and thus strictlyoutperformdecentralization.To represent the planner as a sophisticatedand benevolent mechanismdesigner is clearly extreme. In practice,a planned economy must deal notonly with the problem of individual incentives but also with those resultingfrom communicationcosts, commitmentconstraints,and the incentivesof aself-interestedplanner. The latter incentive problem is all the more impor-tant since centralizedorganizationof economicactivity mplies greatercon-centrationof power. Discretionarycentral direction not only is abused butmayalso induce agents to engage in wasteful"influenceactivities"Milgromand Roberts 1988).Although these problemsare important,we shallconcentrateon only oneof the additional constraintsbesides privateinformation faced by authoritar-ian systems:the planner's imitedabilityto commit.When productiontakes along time, it is often impossiblefor the plannerto be committedtojust whenshe is to intervene and when not. Once she acquires the power to directagents,she mayintervene even when she oughtnot to, especially n an ex antesense. Consequently, ndividualagents'incentivesmaybe adverselyaffected:they may not undertake socially desirable investments for fear of being


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DECENTRALIZATION 823expropriatedex post, or they may not disclosevaluable privateinformationfor fear that the planner might use this information against them. This Ap-pendix illustrates how lack of commitment in our model may prevent theplanner from eliciting privateinformation about firms' costs, and how thisleads to inefficiency.The simplest way of introducing noncommitment into our model is tosuppose that some time elapses betweenthe moment when a firmis selectedfor production and the time when it hascompleted production.Then, follow-ing the literatureon dynamic ncentive contracts see, e.g., LaffontandTirole1988), we shall assumethat the planner cannot commit to a long-term incen-tive scheme, meaning one that covers the entire relationship with the firm.Specifically,consider the followingsequence of moves.1. The planner sets up an initial incentive scheme in which she requireseach firmto report its cost of production;the mechanismspecifieshow firmsare to be selected on the basisof the vectorof costannouncements.Paymentsmay also be made at this point, on the basisof announcements.2. The selected firms then preparefor production,which takesplacein twostages.First, the firmsbuild a plant that costsS. In a second stage, the firmsuse the plantto produce the good at cost c. Thus the total costsof a high-costfirm are given by SH = S + CH; or a low-cost firm, SL = S + CL.3. The initial incentive scheme lasts only until the completionof the firststage of production. It can specify payments only contingenton the plant'sbeing set up (and, of course, contingent on the cost announcementse = (VA;CB)of firms A and B, respectively).We represent the initial incentive scheme by {b#(8);PWl()},where i = A, B,bi(e)is the probability of selecting firm i given e, and P'l(e) is the transfer tofirm i given c, if it is selected and sets up a plant. Without loss of generality, weassume that a firm gets zero if not selected.Between the first and second stages of production the planner must offer anew contract to the firms. If two firms have been selected by the initial incen-tive scheme, then the second incentive scheme is like the first one, but noweach firm has sunk S and the planner knows each firm's initial cost announce-ment. If only one firm has been selected, the second incentive scheme simplyoffers a payment P2(0) for production in the second stage. The firm can, ofcourse, reject the offer, in which case one period goes by before the plannercan make a new offer. In other words, if only one firm has completed the firststage of production, that firm and the planner play a bargaining game withone-sided offers (by the planner) under possibly incomplete information. (Iftwo firms have completed the first stage of production and both firms rejectthe planner's new offer, then again the planner must wait for one periodbefore making another offer, etc.)Finally, to complete the description of our game, we must specify whathappens when the initial incentive scheme is rejected by both firms and whathappens when neither firm is selected initially. In the first case, the plannermust wait one period before offering a new mechanism; similarly in thesecond case, with the difference that when the planner makes a new offer sheknows the initial cost announcements of each firm.We shall assume that the planner's goal is to maximize consumer's surplus.This involves, among other things, minimizing payments to firms. (Alterna-tively, one may assume that the planner wishes to minimize payments to thefirm because such payments involve a [resource] "cost of public funds." SeeCaillaud et al. [1988].) Thus if the planner learns from the initial cost an-nouncements that the selected firm has low costs, she will make a low-price


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824 JOURNAL OF POLITICAL ECONOMYoffer in the future. Anticipatingthis,a low-costfirm maybe tempted initiallyto conceal its low costs, even if it therebyreduces its chancesof being chosen.A complete analysis of this problem is complex. We shall derive only twopropositions that show that theplannercannotavoidselecting n inefficientirm(withpositive robability).hus the comparisonbetween clumsycentralizationand decentralizationemphasized in the text is not too misleading f in fact theproper comparison is between decentralizationand sophisticatedplanningwithout commitment. Our clumsy plannercan be interpreted as a caricatureof a sophisticatedplanner with limited commitment ability.PROPOSITIONAl. Any initial mechanismthat never selects an inefficientfirm is not incentivecompatible.Proof.First,anyinitialmechanism hat never selects an inefficientfirm mustbe a mechanism that induces full separation between the high- and low-costtypes.Startingfrom this observation, et bl(CH; CL)be the probabilityof selectingfirm 1 when the profile of cost announcementsis (CH;CL).We claim that nofully separatingmechanismwith bl(CH; CL) = 0 is incentive compatible.Thisfollowsfrom the incentive-compatibilityonstraints.Consider first the high-cost type. If it announces its true costs, then withprobabilityq its rivalhas low costsand the high-cost firmwill not be selected,since bl(CH; CL) = 0. With probability1 - q, the rivalfirm also announceshigh costs, in which case our firm is selected with probability bl(CH; CH). If it isselectedand setsup a plant,it gets a net payoffPI(CH; CH) - S from the firstincentive scheme, and its continuation payoff is zero because the plannernever offers more than CH in the continuationgame. Suppose now that thehigh-cost type lies and announces low costs. Then its expected payoff fromthe initial mechanism s qbl(CL; CL)[Pl(CL;L) S] + (1 - q)bl(CL; CH)[Pl(CL;CH) - S]. Again,its continuation ayoff s zero.It is not the case (as one mightthink)that the high-costfirmthatlies in the firstperiodwill suffer for it in thesecond period. For it can guarantee itself a continuationpayoff of zero byrefusing to complete production (thatis what Laffont and Tirole [1988] call"grab the money and run"), and it will never get a positive payoff sincetransfers never exceed CH. Thus the high-cost type'sincentiveconstraint sgiven by

(1 - q)bl(CH; CH)[P1(CH; CH) -S (AS' qbl(CL; CL)[Pl(CL; L)- S] + (1 - q)bl(CL; CH)[Pl(CL;H) S].

The low-costtype'sincentive constraint s given byqbl(CL; CL)[PI(CL; CL) - S] + (1 - q)bl(CL; CH)[PI(CL; CH) S]

' (1 - q) bL(CH; CH)[PI(CH; CH) - S] + (CH - CL) (A2)+ [1 - 2b1(CH; CH)] C C

The left-hand side followsfrom the fact that the low-costtype'scontinuationpayoffs are zero under full separation,since the planner will never offer atransfer above CL to what she believes is a low-cost firm. The right-handside is explainedby the fact that if a low-costfirmcheats,it can get an infor-mationalrent of CH - CL if it is selected immediately and of 8(CH - CL) if itis selected one period later. (The firm may be selected one period later ifthe initial mechanismselectsneither firm;this event occurs with probability


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DECENTRALIZATION 8251 - 2b,(CH; CH) ' 0. It is then optimal for the planner to pick one of the[apparently]high-cost firms at random in the next period; thus each firmwill be selected with probability 1/2[1 - b1(CH, CH) - b2(CH, CH)]-)Comparing(15) and (16), we can easily see that they both cannot hold. Inother words, there does not exist a fully separating initial mechanism withbI(CH; CL) = 0. Q.E.D.Remark.-Among other things, proposition Al says that a mechanismthatinitially does not select any firm at all is not incentive compatible.Such amechanism would give zero continuation payoffs to both cost types, and alow-costtype can improve its payoff by lying and getting an informationrentwith positiveprobability.COROLLARY.he first-best solution cannot be implemented by a plannerfacing the commitment constraints described above.

Proof. The first-best requires bl(CH; CL) = 0. QE.D.The commitment constraint not only prevents the planner from imple-menting the first-best but also prevents her from replicating the market solu-tion. For one thing, she cannot commit not to intervene in the future. But,worse, she cannot even replicate the market outcome in the first period sincethis again involves bl(CH; CL) = 0. By proposition Al, this is not incentivecompatible. Becauseof thecommitmentonstraints, heset offeasibleplanning proce-duresdoes not includethe marketmechanism.Emulating decentralization may notbe an option for a central planner.

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826 JOURNAL OF POLITICAL ECONOMYLaffont, Jean-Jacques, and Tirole, Jean. "The Dynamics of Incentive Con-tracts." Econometrica56 (September 1988): 1153-75.Milgrom, Paul, and Roberts, John. "An Economic Approach to InfluenceActivities in Organizations." AmericanJ. Sociology94 supply. 1988): S154-S 179.Milgrom, Paul, and Weber, Robert. "Organizing Production in a Large Econ-omy with Costly Communication." Cowles Foundation Discussion Paperno. 672. New Haven, Conn.: Yale Univ., July 1983.Milward, Alan S. War, Economy, and Society, 1939-1945. Berkeley: Univ.California Press, 1977.Sah, Raaj Kumar, and Stiglitz, Joseph E. "Human Fallibility and EconomicOrganization." A.E.R. Papers and Proc. 75 (May 1985): 292-97.Schelling, Thomas C. The Strategy of Conflict. Cambridge, Mass.: Harvard

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