MIT Sloan School of Management Working Paper 4336-01 May 2001
INDIRECT ADJUSTMENT-COSTS UNDER ALTERNATIVE COORDINATION REGIMES
Birger Wernerfelt
© 2001 by Birger Wernerfelt. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit including © notice is given to the source.
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Indirect Adjustment-Costs under Alternative Coordination Regimes
Birger Wernerfelt*
May 1, 2001
*Professor of Management Science, MIT Sloan School of Management, Cambridge, MA 02142, 617-253-7192,
[email protected]. I am grateful for comments from Oliver Hart, Duncan Simester, Steven Tadelis, Miguel Villas-Boas, and seminar participants at MIT. Of course, the usual disclaimer applies. JEL Codes: D2, L2 Key Words: Theory of the Firm, Coordination, Communication.
Indirect Adjustment-Costs under Alternative Coordination Regimes Abstract The paper is a study of barriers to coordination in terms of agents’ incentives to search for and communicate complementary information. In particular, I look at the value of commitment by comparing game forms in which a contract is negotiated prior to, versus after, search and communication. The comparison depends on three effects. (1) The bargaining power effect: Since the decision to communicate reveals information about preferences, it implies a loss of bargaining power when the players negotiate ex post. This hurts the incentives to communicate and therefore the incentives to search. (2) The incentive transfer effect: If the gains from adjustment accrue unevenly, ex ante negotiation may leave one of the players without incentives to communicate and search. With ex post negotiation, that player can bargain for a share of the gains. (3) The bargaining efficiency effect: The negotiation process itself may be more efficient ex post because more information has been revealed. The net effect depends on the magnitude of the gains and their accrual. If negotiation normally leads to agreement, it is better done ex ante in cases where adjustments yield smaller, more evenly accruing, gains. When the gains are larger and accrue less evenly, ex post negotiation implements more communication and search.
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I. INTRODUCTION Division of labor makes it necessary for agents to coordinate, but at the same time gives them opportunities to acquire different information. The paper is a study of barriers to dynamic coordination, adjustment, in terms of agents’ incentives to search for and communicate complementary information. More specifically, I look at the effects of commitment. When adjustment requires one player to communicate information to the other, should the terms of trade be negotiated before or after communication? The answer depends on the relative magnitudes of three effects. Communication prior to negotiation is (1) -unattractive, because senders prefer their opponent to know as little about their preferences as possible (the bargaining power effect), (2) -attractive, because the ability to negotiate for a share of the gains from an adjustment can give a player incentives to help implement it (the incentive transfer effect), and (3) -attractive, because the revealed information contributes to the efficiency of the negotiation process (the bargaining efficiency effect). Put differently, if bargaining normally leads to agreement, early negotiation is good because players do not worry about loosing bargaining power, and it is bad because payoffs can not be contingent on adjustment. The relative magnitudes of these effects depend on the importance of using payoffs to provide incentives for adjustment, relative to the value of bargaining power due to asymmetric information. The former factor weighs more when non-contractible gains from adjustment are larger and accrue less evenly, and the latter factor weighs more when gains from individual adjustments are smaller and accrue more evenly.
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Application Consider a “marketing” manager, who knows about the kinds of products consumers want, and a “manufacturing” manager, who knows about the kinds of products that can be made. While the two managers know how to make and market an “old” product, they may be able to do better if they can identify a “new” product, which would bring in higher revenues. We assume that the revenues accrue to the marketing manager. The new product may be more costly to produce or market and it may even be infeasible in the sense that it is impossible to make or sell. However, if it is feasible, the two managers can identify it by communicating. Let us first think of a situation in which the manufacturing manager’s salary is fixed ex ante. In this case, his interest in the new product depends only on whether it is harder for him to produce. If it is, he can claim that the new product can not be manufactured. Revenues and marketing costs do not play a role. So an opportunity could be missed because an employee with indispensable information can not be paid to cooperate. Suppose next that the manufacturing manager’s salary is negotiated after he has had a chance to communicate about the new product. In this case, his decision to communicate or not reveals information about his relative costs for the new and the old product. He will be tempted to communicate if the new product is cheaper for him, but since the marketing manager knows this, she will be able to negotiate for a share of his cost savings. So concerns about bargaining power may cause the manufacturing manager to withhold communication even if the new product is cheaper for him. On the other hand, large revenues will affect his incentives to communicate, since he can hope to negotiate for a share of them.
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Plan of the Paper In Section II, I formulate the basic model, introduce the two specific game forms representing ex ante and ex post negotiation. To illustrate the three effects, I compare their ability to implement search and communication in three extreme cases. Some links to the theory of the firm are discussed in Section III. .
II. MODEL AND GENERAL RESULTS Two players, A and B, she and he, may cooperate on an “old” project or a “new” project. The essential feature of the model is that the new project only is feasible if (i) both players search for and find it, (ii) one player communicates his finding to the other, and (iii) the latter selects the new project over the old. The only ex ante difference between the players is that prior to side-payments, a fraction ra of the revenues received for the project accrue to A, while B gets (1- ra). We will initially let B decide whether or not to communicate and if he does, give A the choice between the two projects. In the Appendix, I show that the results are robust to reversal of the roles. A project is defined by two tasks that the players have to carry out to implement it. The first of these tasks has to be carried out by A and the second has to be carried out by B. The players are both indispensable in the sense that both are necessary to implement any project. Neither has an outside option other than doing nothing, and receiving zero. In order to focus on cooperation conceived as information exchange, I have chosen to abstract from moral hazard in the determination of productive efforts. That is, I am effectively assuming that the players’ levels of effort are unaffected by the game form governing the relationship. This is probably not a realistic assumption. It seems plausible that higher
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effort levels can be enforced when the players negotiate with better knowledge of their tasks. For purposes of the following, this would imply an upward shift in the efficiency of the game form with ex post negotiation. However, the qualitative results of the analysis would not change. The feasibility of the old project is guaranteed, and the nature of the tasks is known. However, the costs of the tasks, Coa for player A, and Cob for player B, are the private information of those players. We assume that Coa and Cob are i.i.d. draws from a distribution F over the unit interval. It will initially be convenient to assume that F is symmetric around ½ and has no mass points. The revenues of the old project are Ro. The players gather information under moral hazard. Specifically, if a player incurs a positive search cost s, he or she finds the new project with probDELOLW\ ,WLVLPSRUWDQW WKDW VLQFHRWKHUZLVHDGHFLVLRQQRWWRFRPPXQLFDWHEHFRPHVDVLQIRUPDWLYHDVD decision to communicate.) Without search, there is no chance of finding the new project. (To keep things simple, I assume that it is not observable whether a player searches.) The new project is only feasible only if both players find it. In that case its costs, Cna and Cnb, are i.i.d. draws from the same distribution as those of the old project. New projects are more attractive in expectation because their revenue Rn5o. In addition, a new project offers a fresh cost draw. If a new project is found, the players will learn from information WKDWWKH\UHFHLYHWKURXJKWZRVLJQDOVWKDWZHZLOOODEHO DQG 7KHVLJQDO KDVWZR components. It reveals Cna, and it contains partial information about the tasks that define WKHQHZSURMHFW:HODEHOWKLVODWWHUFRPSRQHQW DQGDVVXPHWKDWLWFDQEH communicated, but does not reveal any information about Cna7KHVLJQDO PD\RQO\EH received by player A.7KHVLJQDO ZKLFKPDy be received by B only, has a similar
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structure. It reveals Cnb,DQGLWFRQWDLQVSDUWLDOLQIRUPDWLRQ DERXWWKHWDVNVWKDWGHILQHWKH new project./LNH FDQEHFRPPXQLFDWHGDQGLWGRHVQRWUHYHDODQ\LQIRUPDWLRQDERXW Cnb. The tasks that define the new project are unforeseen ex ante and to learn the nature of these tasks a player needs to know both DQG 7KHSOD\HUVPD\FRPPXQLFDWH DQG WRHDFKRWKHU and the truthfulness of communication is observable, but not verifiable. So while the receiver of a message can tell whether it is true or not, the parties cannot contract on the veracity of communication. Without this critical assumption, contracts can trivially solve the problem. Fortunately, this “softness” of information (see Tirole, 1986) is a very natural assumption in the context of division of labor. (Think of a pair of marketing and manufacturing managers or two coauthors.) If only ANQRZV RQO\BNQRZV DQGRQHFDQQRWPDNHVHQVHRIRQHZLWKRXW the other, then it is hard to see how a third party can rule on claims about either. The fact that receipt of messages is non-verifiable prevents the players from agreeing to bilateral communication, thus implementing ex post symmetric and complete knowledge of payoffs. I take this contractability argument even further and assume that outsiders understand so little about the projects, that contracts cannot depend on whether an implemented project is new or old. Alternatively, I assume that we focus the analysis on those projects for which this is true. This implies that the players can do no better than to negotiate a contract over w, a transfer from A to B if a project is implemented. (A somewhat related argument is made by Stein, 2000). 6LQFHRQO\RQHSOD\HUQHHGVWRNQRZ ¶DQG ¶ZHDVVXPHWKDWRQO\RQHSOD\HU may communicate. If there is communication, the receiving player can choose between the new and the old project. In this context, it is important that the costs of the old project are
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private information. Without this assumption, communication may be discouraged simply because bargaining over the old project may be more efficient. More subtly, it serves to encourage communication because a player reveals information about the costs of the old project by choosing it after receiving communication about the new project. So the communicating player gets some information in return for what he or she gives up by communicating. It is assumed that communication reveals information only in the weakest possible way. It may be reasonable to assume that the content of communication reveals something about costs. However, I make the weaker assumption that no information is revealed beyond that contained in the choice to communicate. This conservative assumption means that communication results in a rather indirect loss of bargaining power. In modeling the effect of information on bargaining power, I have two concerns. First, I would like to make rather weak assumptions about how power varies with informational asymmetries. Secondly, I want to separate the distributive effects from the pure efficiency effects of information. To accomplish this, I assume that bargaining, independently of when it takes place, consists of a single take-it-or-leave-it offer that is made by A with probability p, and else by B. I model p as a function of two arguments, “how little B knows about A’s costs”, and “how little A knows about B’s costs”. One could think of these as posterior probability distributions that are ordered by some measure of dispersion. However, in the following, I will be looking at situations in which the arguments are identical, plus a case in which a single real number can serve as a sufficient statistic for how much is known about a player. So the analysis is consistent with a lot of ways of measuring these arguments. For now, it suffices to say that p
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(i)
-is increasing in the first argument and decreasing in the second,
(ii)
-satisfies the symmetry condition p(x,y)=1-p(y,x), and
(iii)
-is linear in probabilities such that if BDVVLJQVSUREDELOLWLHV - WRVWDWHVDQG in which he knows x1 and x2, respectively, about A’s costs, and x1 and x2 are nested in the sense that one could be a posterior resulting from the other, then S
S [1,y)+(1- S [2,y), and similarly for A’s beliefs about B’s costs.
The second condition implies that p(x,x)=1/2. The third condition could be weakened considerably, but has the advantage of making the analysis notationally much more transparent. It is unpleasantly strong and is inconsistent with many reasonable measures of the arguments of p. However, as we will see, it is much less offensive in the special cases we will be looking at. The negotiation mechanism itself is clearly arbitrary and, in general, less than second best. For purposes of the analysis, its two most critical properties are (1) that payoffs responds to informational advantages, and (2) that bargaining becomes more efficient when the players have better information. These are properties are satisfied by most reasonable models of bargaining, such as the sealed bid model of Chatterjee and Samuelson (1983). Let me now define two game forms. One in which the players commit to a contract negotiated before search and communication, and one in which there is ex post negotiation. It is clearly possible to look at a number of other game forms. In particular, one could imagine a game form with some intermediate degree of commitment (Aghion and Tirole, 1997; Rogoff, 1985) to an ex ante negotiated contract. However, as a first cut, I will focus on a comparative analysis of the two extreme cases with full and no commitment.
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The “Ex Ante Negotiation” and “Ex Post Negotiation” Game Forms Because either player can be asked to communicate, there are two versions of each of these game forms. I will show in the Appendix that their performances are identical. In this Section I will assume that B communicates, such that A gets to select the project.
The “ex ante negotiation” game form is defined by the following six stages: EA1. The parties write a contract, which says that B will carry out an unspecified task at A’s request. In return, B gets the unconditional transfer t, as well as w if the project is implemented. The players set w to maximize joint payoffs, and negotiate t according to the bargaining mechanism described above. EA2. The players may invest s in search. EA3. The parties learn their costs for the old project and may find the new project and thus DQG UHVSHFWLYHO\ EA4. B may communicate to A. EA5. A may ask B to perform a specific task (or the game will end). EA6. The players may carry out their tasks, and if so, payoffs are distributed. (Otherwise, there are no payoffs.)
The “ex post negotiation” game form is defined by the following six stages: EP1. Each player may expend search costs s. EP2. The players learn their costs for the old project and may find the new project and WKXV DQG UHVSHFWLYHO\ EP3. B PD\FRPPXQLFDWH WRA.
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EP4. A announces whether she wants them to carry out the old or the new project. The ODWWHUEHLQJIHDVLEOHRQO\LIVKHNQRZVERWK DQG EP5. The parties may write a contract, which says that B will get w to carry out the task associated with the chosen project. They then determine w according to the bargaining mechanism described above. EP6. If agreement is reached, the players carry out their tasks, and payoffs are distributed.
We will compare these two game forms under several extreme circumstances.
1. Rn=Ro and Ro large: Ex Ante Negotiation is Best. In this subsection, we will isolate the bargaining power effect by looking at a case in which the revenues are identical, but very large. This means that (i) preferences between projects are only based on costs, and even the largest cost differences are swamped by minor variations in bargaining power, and (ii) all bargaining is efficient. (The latter implication requires that, given the “fatness” of F’s upper tail, revenues are so large that expected payoffs are maximized by making offers that are accepted with probability one.) With ex ante negotiation, communication can not influence bargaining power, so the equilibrium is very simple. Given a choice between the old project and a new project, A will simply select whatever is cheaper for her, and B will communicate whenever the new project is cheaper for him. So in the end, ¼ of all new projects are implemented, accounting for ½ of those that ideally should be implemented.
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Result 1EA: If Rn=Ro, Ro is large, and s< 2>]-z(1-F)]dF, the game form with ex ante negotiation has an equilibrium in which both players search. B communicates if and only if Cnb &ob, and if he does, A will select the new project if and only if Cna &oa.
Proof: We will analyze the game backwards. In stage 5, A will continue if her payoffs, raRo -w, are greater than her costs for the project. Given a choice between the old project and a new project, A will simply select whatever is cheaper for her. So if she has the option, A will select the new project with probability 1/2. In stage 4, if B has received KHZLOO communicate whenever the new project is cheaper for him, or with probability 1/2. Assuming that both players search, we first look at A’s incentives in stage 2. Her expected payoff consists of three terms. They represent her surplus (i) if B does not communicate or A does not find the new project, (ii) if B communicates, and A decides not to ask for the new project, and (iii) if they implement the new project. In case (i) A’s expected costs are zdF, in case (ii) they are z(1-F)2dF, and in case (iii) they are z(1F)2dF. Her expected payoff reduces to [1- 2+(1/2) 2][raRo-w-zdF]+(1/2)(1/2) 2[raRo-w-z(1-F)2dF]+ (1/2)(1/2) 2[raRn-w-z(1-F)2dF]-s = raRo-w- 2>]z(1-F)]dF -s.
(1)
If A does not search, her expected payoffs are raRo-zdF-w. So if B searches, A will do so if V 2>z/2-z(1-F)]dF. Turning now to B, his payoff from searching when both players do so, is given by (1- 2)[(1-ra)Ro-w-]G)@ 1/2) 2[(1-ra)Ro+w-z(1-F)2dF]+ (1/4) 2[(1-ra)Ro+wzF2dF]+(1/4) 2[(1-ra)Ro+w-z(1-F)2dF]-s =
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(2)
(1- ra)Ro +w- 2[z/2+z(1-F)]dF-(1- 2)]G)-s.
(3)
So in this game form B’s search incentives are exactly as A’s. There is an equilibrium, in which both players will search if V 2>z/2-z(1-F)]dF.
(4)
Finally, in stage 1, if the players set w=(ra-1/2)Ro, both of them will want to continue at all later stages. Q.E.D.
The game form with ex post negotiation is much more complicated. However, the analysis is simpler when we can assume that all offers are accepted. It is further simplified if revenues are very large, such that concerns for bargaining power dominates preferences based on cost differences between projects. Under these circumstances, it will turn out that there is no communication and thus no search in equilibrium. The reason is that A can exploit B’s communication strategy to select a bargaining scenario in which she has more power. If only B-types with very low Cnb and very high Cob communicate, all A-types will select the new project and thus be in a superior informational position. Conversely, if most B-types cummunicate, all A-types will select the old project and thus force bargaining over the old project knowing a lot about Cob. As a result, the only equilibrium is one in which no B-types communicate and no new projects are implemented.
Result 1EP: If Rn=Ro and Ro is large, the game form with ex post negotiation has no equilibria in which B communicates. There is thus no search either.
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Proof: We first need a bit of notation. The key components of a candidate equilibrium are a communication strategy for B and a selection strategy for A. Any pair of such strategies, combined with a pair of choices, give each player a posterior probability distribution over the costs of the other. There are five relevant scenarios: (1) B communicates and A selects the new project, (2) B communicates and A does not select the new project because she prefers not to, (3) B communicates and A does not select the new project because she did not find it, (4) B does not communicate because he prefers not to, and (5) B does not communicate because he did not find the new project. If both players know the scenario, we use (xi,yi), i=1,2...5, to denote the arguments of p in these five scenarios. Furthermore, we let (Cna1,Cnb1) denote the highest costs A and B possibly could have in scenario 1, while (Coai, Cobi), i=2,3,4,5, denote the highest possible costs in the other scenarios. Because B can not distinguished scenarios 2 and 3, x2=x3, and because A can not distinguished scenarios 4 and 5, y4=y5. The costs satisfy similar conditions. Whoever makes the offer at the bargaining in stage 5 will take almost all the surplus, leaving the other player just enough to cover the highest cost she or he possibly could have. So for example in scenario 1, A will bid wa= Cnb1-(1-ra)Rn and B will bid wb= raRn -Cnb1. Given that revenues are so high that all offers are accepted, when A compares scenarios (1) and (2) in stage 4, she will select the former if p(x1,y1)(Rn -Cnb1-Cna)+[1- p(x1,y1)](Cna1-Cna)> [ S(x2,y2)+(1- S(x3,y3)] (Ro–Cob2-Coa)+[1- S(x2,y2)-(1- S(x3,y3)](Coa2-Coa). (5) This can always be written in the form
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Cna [ S [4,y4)+(1- S [5,y5)](Cob4-Cob)+ [1- S x4,y4)-(1- S [5,y5)](Ro –Coa4-Cob). (7) This can always be written in the form Cnb -7 a)]p(- a, b)+(1- S b) For intermediate values of
a,
(9) involves a gamble (with probabilities T and 1-T) between
two inferior options. This is dominated by taking the higher p for sure, either by such that T( a)=0, or by
a=1,
(9)
a=-1,
such that T( a)=1. Intuitively, A does not want to separate.
Given this, we go back to stage 3 of the game and look at the probability that B gets to make the offer. If
a=1
and T( a)=1, B will make the offer with probability
7 b)[1-p(1 b)] >-7 b)][1-p(1,- b)]+(1-
16
(10)
Just like (9), (10) involves a gamble between two inferior options for intermediate values of
b
(with probabilities T and 1-T). This is dominated by taking the higher p for sure,
either by
b=-1,
such that T( b)=0, or by
b=1,
such that T( b)=1.Like A, B does not want to
separate. Similarly, when
a=-1,
T( a)=0, B will make the offer with probability
7 b)[1-p(1 b)] >-7 b)][1-p(1,- b)]+(1- which is identical to (10). This leaves us with 1 and –1 as the only candidate values of
(11) b.
Given that the search costs s are positive, neither player will search, and there will be no communication. Q.E.D.
Summarizing Results 1EA and 1EP, we see that ex ante negotiation implements efficient new projects half the time, while none are implemented under ex post negotiation. The bargaining power effect favors ex ante negotiation.
2. Rn>>Ro, Ro large, and ra=1: Ex Post Negotiation is Best. To illustrate the incentive transfer effect, we now look at another extreme situation. We assume that Rn is much larger than Ro, and that all revenues accrue to A. This means that it is critical to implement the new project, but that B has no direct incentives to do so. (If ra 2z(1-F)2-(1- 2)z]dF-s. (14) If B does not search, he can expect w-]G). So given that A searches, B will do so if V 2] )-1)dF.
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(15)
Since Rn- Ro is large, we see that B’s search incentives are much weaker than A’s. Therefore, both players will search if (15) holds, and neither player will search if this does not. Q.E.D.
In this case the game form with ex post negotiation is much simpler.
Result 2EP: If Rn is much larger than Ro, Ro is large, ra=1, and s< 2(Rn-Ro)/2, the game form with ex post negotiation has an equilibrium in which both players search. B always communicates and A always selects the new project.
Proof: In the postulated equilibrium neither party learns anything about the costs of the other. So the prior beliefs never updated and p=1/2. Appealing to the size of revenues, as in the proof of Result 1EP, we disregard costs as well as payoffs when the other player makes the offer. So if A has a choice, she expects Rn/2 from the new project and Ro/2 from the old project. She will therefore always choose the new project in stage 4. Going to stage 3, if B communicates, he will be able to negotiate for revenues Rn/2 with probability DQGRo/2 with probability 1- ,IKHGRHVQRWFRPPXQLFDWHKHFDQH[SHFWRo/2. So consistent with the hypothesized equilibrium, all B types will communicate and all A types will select the new project. In this game form B can expect to bargain for a share of Rn, and his communication strategy is therefore responsive to revenues even though ra=1. The bound on search costs finally follows from the facts that each player can expect Ro 2(Rn-Ro)/2s if they search, and Ro/2 if they do not. Q.E.D.
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Comparing Results 2EA and 2EP, we see that ex post negotiation always implements the new project, while ex ante negotiation only implements it half the time. The incentive transfer effect favors ex post negotiation.
3. Rn=Ro and Ro small:Ex Ante Negotiation May Be Best. We now illustrate the bargaining efficiency effect. In the previous subsections, we have assumed that Ro is so large that all bargaining processes are efficient. It is never an equilibrium for a player to make a low bid and risk not trading. We will now change this and see how the information revealed by communication can increase the efficiency of the bargaining process. If the bargaining efficiency effect is very small, we saw in Results 1EA and 1EP, that ex ante negotiation will still be best if Rn=Ro. However, we will here show by example that the bargaining efficiency effect can be so large that ex post negotiation is best, even with a small bargaining power effect working against it. To this end, we will completely neutralize concerns for bargaining power by fixing a constant p=1/2, and appeal to continuity to argue that the results continue to hold for “almost” constant p-functions. Since we are just looking for an example, we assume that costs are drawn from a simple multinomial distribution F’, that assigns equal probabilities to costs being C1 2 &2 - &3 RU&4 where
Result 3EA: If Ro= Rn
Ua FRVWVDUHGUDZQIURP)¶DQGV WKHJDPH
form with ex ante negotiation has an equilibrium in which both players search. B will communicate if and only if Cnb equals C1 or C2, and if he does, A will select the new project
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if and only if Cna equals C1 or C2 unless (Cna , Coa)=( C2 , C1). The players only trade if both their costs are C1 or C2.
Proof: Suppose that w=1/2. In this case players only want to trade if their costs are C1 or C2, and the communication and selection strategies are obvious except for the claim that B will communicate when (Cnb , Cob)=( C2 , C1). If this B type does not communicate, he ZLOOWUDGHZLWKSUREDELOLW\òDQGJDLQ LIKHGRHV+RZHYHULIKHFRPPXQLFDWHVKHZLOO WUDGHWKHQHZSURMHFWZLWKSUREDELOLW\JDLQLQJ DQGWUDGHWKHROGSURMHFWZLWK SUREDELOLW\JDLQLQJ 6RWKHH[SHFWHGUHWXUQIURPWKHODWWHUis higher and the claim is true. If we mechanically calculate the players’ payoffs, we find that A and B can expect DQG UHVSHFWLYHO\IRUDWRWDOVXUSOXVRI 6LQFHWKH\HDFKFDQ H[SHFW LIWKHUHLVQRVHDUFKLWLVQHFHVVDU\What s
