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Repeated contract negotiations with private information
ELSEVIER
Japan and the World Economy 7 (1995) 447-472
/" Z a n and the WORLD ECONOMY
Repeated contract negotiations with private information John Kennan* Department of Economics. University of Wisconsin, Social Science Building, 1180 Observatory Drive, Madison, WI 53706, USA
Received April 1994; accepted December 1994
Abstract Labor contracts are repeated, and the current negotiation apparently affects negotiations on the next contract. There is evidence that the likelihood of a strike is influenced by what happened before the previous contract was signed. To analyze whether private information might explain this, a model of negotiations between a buyer and a seller is developed where the "pie" in each contract follows a Markov chain, with transitions observed privately by the buyer. Each negotiation is a sequence of offers by the seller; the buyer can only accept or reject each offer, and a contract is signed when the buyer accepts. Keywords: Bargaining; Private information; Labor contracts; Strikes JEL classification: C78; D82; J41; J51
1. Introduction M a n y prices are negotiated in the context of repeated bargaining relationships, with an element of bilateral m o n o p o l y involved. The leading example is the price of labor services, set in union contracts that are renegotiated every few years; other examples include contracts for coal, steel and other raw or intermediate inputs. Uncertainty about the other party's reservation price is apparently a standard feature of these negotiations. * Tel.: 608-262-5393. Fax: 608-263-3876 or 608-262-2033. E-mail:jkennan(b~macc.wisc.edu. 0922-1425/95/$09.50 © 1995 -Elsevier Science B.V. All rights reserved SSD! 0922-1425(95)00020-8
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Recent work on labor contracts has emphasized the possibility of explaining collective bargaining outcomes in terms of the incentive-compatibility or truthtelling constraints arising when either party to a bargaining game is endowed with unverifiable private information. 1 The central point in this work is that outcomes such as strikes that appear inefficient can be seen as a necessary cost of extracting information. Similarly, theoretical analyses of macroeconomic fluctuations have emphasized the possibility that labor contracts might prescribe layoffs as an efficient response t ° an employer's privately observed signals about the value of labor's product, even though the laid off workers' marginal product exceeds their reservation wage. The empirical literature on labor contracts has shown convincingly that the outcome of the current negotiation is substantially influenced by what happened when the previous contract was negotiated. 2 This paper develops a model of repeated negotiations with private information on the buyer's side that can be used to confront some of the regularities found in the labor contract data. The buyer's valuation changes according to a two-state Markov chain, and in equilibrium the seller goes through cycles of optimism and pessimism, triggered by whether aggressive bargaining tactics are successful in each negotiation. In the labor context the union calls a strike periodically in order to update its information about the employer's valuation, and a long strike with a low wage settlement at the end induces the union to lie low in the next few contract negotiations. In order to obtain concrete results it is necessary to adopt a specific set of bargaining rules, and a stochastic process for the buyer's valuation. Here it is assumed that the uninformed seller makes offers that the buyer must either accept or reject, so the equilibrium involves screening or price discrimination. A twostate Markov chain is the simplest interesting process that allows the buyer's valuation to have both transient and permanent components. At one extreme, no transitions occur so that the high-valuation buyer is very wary of revealing its type, because of the "ratchet effect": once the high valuation is revealed, the seller will claim the entire rent in all future negotiations. At the other extreme, the current valuation is entirely transitory, so that the model reduces to a repeated sequence of one-shot screening negotiations. Between these extremes the model is capable of generating equilibria in which the seller makes either a pooling offer or a sequence of screening offers in any given contract negotiation, depending in part on what happened in the previous negotiation. It is assumed throughout this paper that contract duration is given, and that the Markov chain makes one transition for each contract. The literature contains no good theory of contract duration: we do not know why unions 1 See, for example, Hayes (1984), Fudenberg et al. (1985), Kennan (1986), Hart and Tirole (1988), Hart (1989), Kennan and Wilson (1989, 1990, 1993), Card (1990a, b), and Cramton and Tracy (1992, 1994a, b). 2 See, for example, Riddell (1979, 1980), Card (1988, 1990b), and Ingram et al. (1991).
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and employers sign contracts in the first place, and we do not know why these contracts often last three years, sometimes one, two or four years, but rarely longer) There is a possibility that the model developed here might shed light on this question, by linking the contract duration to the degree of persistence in the buyer's private information. In order to explore this one must first analyze the equilibrium for a given contract duration and a given degree of persistence, and then consider how the equilibrium changes as these parameters are varied. The model developed here is related to earlier work by Blume (1990) and Vincent (1990) on the effects of new information that arrives while a bargaining game is in progress. Blume (1990) analyzed a two-type model where the low type can temporarily assume the valuation of the high type, and emphasized that even if the informed party is limited to accepting or rejecting offers made by the uninformed party, there is an important signaling aspect of the negotiations. The model in this paper differs from that of Blume's in two respects: the information structure is a two-state Markov chain, so that both types might change valuations, and the game involves repeated contract negotiations, as opposed to a final sale.
2. Multiple negotiations A serious limitation of the recent literature applying strategic bargaining models to labor negotiations is that each contract is treated in isolation. In practice, labor negotiations involve contracts covering several years, and the parties seem mindful of how the outcome of the current contract will affect negotiations on the next contract. A natural theoretical interpretation is that private information is to some extent temporary, but also to some extent permanent, so that if either side lets the cat out of the bag in the current negotiation, its bargaining position will be weaker when the next contract is negotiated. Recent evidence developed by Card (1988, 1990b) indicates that the likelihood of a strike is heavily influenced by whether a strike occurred before the previous contract was signed, and, if so, by the duration of the strike (see Table 1). Similarly, Ingram, et al. (1991) found that the likelihood of a strike in British wage negotiations is strongly influenced by what happened in the previous year's negotiations: curiously, a "go slow" last year means that a strike is less likely this year, while an overtime ban last year has the opposite effect. At first sight it seems odd that short and long strikes last time would have opposite effects on the strike probability this time. But a private-information 3See, for example, Fischer (1977),Gray (1978), Harris and Holmstrom (1987),Joskow (1987)and Borland and Tracy (1988).
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Table 1 Outcomes of successive contract negotiations Canada
US Current strike duration
Currentstrike duration
Previous strike duration
Total
None
Short
Long
Total
None
Short
Long
None
2198 82
Long
235
Total
2515
67 3.0% 7 8.5% 5 2.1% 79 3.1%
174 7.9% 25 30.5% 29 12.3% 228 9.1%
3651
Short
1957 89.0% 50 61.0% 201 85.5% 2208 87.8%
3105 85.1% 136 61.3% 411 78.7% 3652 83.1%
179 4.9% 35 15.8% 26 5.0% 240 5.5%
367 10.1% 51 23.0% 85 16.3% 503 11.4%
222 522 4395
Notes: The US results are derived from the data used by Vroman (1989); similar results appeared in Card (1988). The results for Canada are derived from a dataset assembled by Harrison (1993), using contract and strike data files from Labor Canada. A short strike means that the duration was 1-14 days (1-10 working days in the Canadian Data).
m o d e l m i g h t e x p l a i n this n o n m o n o t o n i c effect as follows. A s h o r t strike last time m i g h t m e a n t h a t the firm was f o u n d to be rich, a n d is p r o b a b l y still rich, so the u n i o n m a k e s an aggressive d e m a n d which will cause a strike if the firm's c u r r e n t profits d o n o t meet expectations. A peaceful settlement last time m i g h t m e a n t h a t the u n i o n was n o t sufficiently o p t i m i s t i c to t h r e a t e n a strike, a n d thus no new i n f o r m a t i o n was revealed. F i n a l l y , a l o n g strike last time c o u l d m e a n t h a t the firm was f o u n d to be p o o r , a n d so the u n i o n d o e s n o t c o n s i d e r it w o r t h w h i l e to t h r e a t e n a strike now.
3. An Infinite horizon Markov model of repeated negotiations C o n s i d e r a n infinite sequence of c o n t r a c t n e g o t i a t i o n s b e t w e e n a b u y e r a n d a seller where the rent to be d i v i d e d in each T - p e r i o d c o n t r a c t follows a simple M a r k o v process, with t r a n s i t i o n s t h a t are o b s e r v e d p r i v a t e l y by the buyer. B o t h sides m a x i m i z e the p r e s e n t value of expected income, with a c o m m o n d i s c o u n t factor 3 p e r period. F o r e x a m p l e , if v is the present value of a firm's r e v e n u e d u r i n g the t e r m of a l a b o r contract, net of all n o n l a b o r costs, a n d if Wo is the highest w a g e a v a i l a b l e to w o r k e r s d u r i n g a strike, then the rent is v - w o. A s s u m e t h a t the rent follows a t w o - s t a t e M a r k o v c h a i n with c o n t i n u a t i o n p r o b a b i l i t i e s p a n d a over the length of a c o n t r a c t , so t h a t if the rent is low n o w (v = VL), it will a g a i n be low T p e r i o d s hence with p r o b a b i l i t y p, a n d if the rent is high n o w (v = vR) it will be
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high again next time with probability a. 4 It is convenient to use w 0 as the origin and vH - v L as the unit, so relabel 0 = (vL -Wo)/(v n - V L ) as the low rent, with 1 + 0 as the high rent and zero as the seller's opportunity cost. Let A = fix be the discount factor from the beginning to the end of a contract. Then the model is summarized by the five parameters (p, a, 0,6,A). For empirical purposes, A m a y be treated as approximately observable, in the sense that it must correspond to some market discount factor. On the other hand 6 is the discount factor covering one bargaining period, defined as the length of time between offers in each contract negotiation, and this will be treated as unobservable. In the absence of any historical information, the probability of the low valuation is that implied by the stationary distribution of the M a r k o v chain, i.e. 1 -a
Z*=l--tx+l-p
1 -a
1--~'
(1)
where ~b = p + a - 1 measures the degree of persistence in the M a r k o v chain. Let p(s) denote the probability that the M a r k o v chain is in the low state after s transitions, given that the current state is low, and let a(s) be the corresponding probability for the high state. Then p(s) = z* + eSz*,
,~(s) = z~, + 4~Sz~.
(2)
The parameter @governs the extent to which successive contract negotiations are linked. Assume ~b/> O, so that the probabilities p(s) and a(s) do not oscillate. If (~ = 0 information is completely transitory, so that any inference that the seller might draw from the current contract negotitation will be irrelevant by the time the next contract is negotiated. At the other extreme, if ~b = 1 the current information is entirely permanent. Under the interpretation that @ summarizes multiple transitions during the term of a contract, with ~b = @T, the linkage across contracts is made weaker if the contract length is increased, for a given value of ~bo. In the most interesting parameter configuration, the seller would not find it optimal to make a pooling offer unless there is recent information indicating that the buyer is likely to be p o o r this time. On the other hand, if the seller inferred from the previous outcome that the rent was low, it may be optimal to make a pooling offer in the current negotiation (i.e. to make an offer which all types of the buyer will surely accept). This means that the seller learns nothing new in the current negotiation, so when the next contract comes up, the seller's information will be weaker than it is now. If the information process is highly persistent, 4 This notation may be taken as a summary of multiple transitions in the Markov chain during the term of the contract: for instance if the chain makes one transition per period, and the basic transition matrix is Ao, then the transition matrix from one contract to the next is A = Aov, and p and a may be interpreted as the diagonal elements of this matrix. Under this interpretation vL and vn denote expected present values over the life of the contract, given the state in the initial period.
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pooling may still be optimal next time, but eventually the seller's information erodes to the point where it again becomes optimal to screen. This provides an alternative interpretation of the following passage from Hicks (1932): "Weapons grow rusty if unused... The most able Trade Union leadership will embark on strikes occasionally, not so much to secure greater gains upon that occasion.., but in order to keep their weapon burnished for future use..." (p. 146).
In the screening model it is the union's information rather than its strike weapon which grows rusty, but the effect is similar. A Markov-Perfect equilibrium of this game is a renewal process based on the outcome of screening offers made by the seller. Thus if the buyer accepts an offer revealing that the rent is currently high, the continuation game is the same as it was the last time such a revelation was made, and similarly if the buyer rejects sufficiently many offers to convince the seller that the rent is currently low. In each contract negotiation there are two possibilities from the seller's point of view. If information is sufficiently persistent (d~ is relatively high) and if the seller has inferred from a recent negotiation that the rent was low, it will be optimal to make a pooling offer. Alternatively, if the seller believes that the high-rent state is sufficiently likely, a sequence of screening offers will be worthwhile; these offers will be acceptable to the buyer if the rent is currently high, and unacceptable if the rent is low. If the buyer accepts a screening offer, the seller will infer that the rent is high, and so the seller will screen again when the next contract is negotiated (unless perpetual pooling is optimal). Of course the buyer knows that acceptance of a screening offer weakens its bargaining position next time, so the offer must be sufficiently generous to compensate for this. If the buyer rejects all screening offers, on the other hand, the seller infers that the rent is currently low, and it may then be optimal to make a pooling offer next time, and perhaps again the time after that, and so on. A key feature of the equilibrium is the number of pooling offers, K - 1, made by the seller in the sequence of contracts following rejection of a screening offer. The equilibrium is derived by considering the continuation values of the game for the buyer and for the seller, in the various circumstances that may prevail when the seller screens. Screening negotiations last through M offers, and then a T-period contract is signed; let vH, vL and w represent present values of T-period streams as of the signing date. After rejection of M - 1 offers the seller concludes that the rent is currently low, and the seller then makes pooling offers in the next K - 1 negotiations, followed by a screen in the Kth negotiation: call this a "soft" screen, and let U a be the seller's continuation value in this case. If the buyer accepts an offer revealing a high rent now, the seller screens again when the next contract comes up: call this a "hard" screen, with continuation value U b for the seller. This is sketched in Fig. 1, which represents varying degrees of pessimism for the seller. At one extreme, the seller believes the buyer is in the low state now
J. Kennan/Japan and the World Economy 7 (1995) 447-472 Hard
Soft
Screen
Screem
453
Pool
Screen
¢, m
1-¢~
z~
o(K)
~*
P
Fig. 1. The screening and pooling cycle.
with probability p, because there was a screen in the previous negotiation and the buyer was found to be poor. In this situation the seller pools now, and pools again in K - 1 successive negotiations until the probability of the low type has decayed past the screening threshold (*. At this point there is a screen, but the screen is "soft" in the sense that the seller is still relatively pessimistic: the probability of the low state is p(K), which is above the stationary value z~, and so the buyer will not have to reject many offers before the seller gives in. At the other extreme, the seller is most optimistic after the buyer was found to be rich in the previous negotiation; then the probability of the low state now is only 1 - a, so the seller's posture is "hard" in the sense that it will persist through relatively many rejected offers before giving in with a pooling offer.
4. Equilibrium An equilibrium in this model is a reproducible set of continuation values for the buyer and the seller at the point of signing a contract. That is, if both sides take these continuation values as given for all contracts after the current one, and bargain optimally now, then the continuation values at the end of the current negotiation must reproduce the equilibrium continuation values. The equilibrium will be constructed in three steps. First, the sequence of offers that would be acceptable to the high-type buyer is derived from the condition that each of these offers must give the high-type seller the same continuation value as would be obtained by refusing all but the last offer, and causing the seller to infer (wrongly) that the current state is low. Secondly, the seller must determine how much weight to put on each of these offers, subject to the constraint that sequence of remaining offers must be credible after each offer is
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rejected. Finally, the state-contingent continuation values thus calculated must reproduce themselves. In a screening negotiation the seller makes a sequence of offers such that the high-type buyer is indifferent between accepting each offer and waiting for the next one. In equilibrium the high type randomizes so that after M - 1 offers have been made, all high types have accepted, and the seller then offers 0, which is accepted by the low-type buyer. Define VH as the continuation value of the game for a high-type buyer that has just signed a contract at 0, by imitating the low type. During the countdown from the seller's initial offer, labeled wM_ 1' to the last offer wo = 0, the buyer must be indifferent between the last offer and the offer wi that is made when there are i periods left in the countdown. Thus
6iVn=l+O-wi+Adb~'n,
i>l,
Vn - [1 - tr] VL +
(1 -
where d b = screen.
6ub- 1
a V H = V n --
p ) ( V n --
VL),
(3)
is the discount factor from the beginning to the end of a hard
4.1. O p t i m a l s c r e e n i n g
Define U a and U ~ as the seller's continuation values at the start of future soft and hard screening negotiations, which are taken as given when the seller constructs an optimal sequence of offers for the current contract. Let u~ be the seller's continuation value if the buyer accepts w~ in the current negotiation. Then
ui=-wi+AUb=I+O+Adb~'H-OIVH+AU b, i>_l
(4)
and 1- A r u o = O~_A+
AKua"
(5)
Suppose that at the start of the current negotiation the probability that the buyer's valuation is low is (, according to the seller's information. Then if Pi is the probability that the buyer accepts w~, with Po = (, the seller's valuation is M - 1
M-
U= 2 tSM-l-ipltti=d ~ i=0
1 1J~i''i
t$i'
(6)
i=0
where d = 6 M- 1 is the discount factor from the beginning to the end of the sequence of offers. Since the contingent valuations u i depend only on how the
J. K e n n a n / J a p a n and the W o r l d Economy 7 ( 1 9 9 5 ) 4 4 7 - 4 7 2
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game continues after the current negotiation ends, the seller must take them as given, and maximize by choosing the probability sequence {Pi}. Everything the seller needs to know about future contracts is summarized by the screening threshold (* and the pair (7, 70) defined by 1 + 0 + A U b + Adb~"n
VH
~' -
Yo- 1 +
ui
- VH t- '~'
0(1-AX)/(1-A)+AKU "
v.
uo = - - + 1.
v.
(7)
Here Vis the ratio of the joint continuation value to the buyer's value, given that the current state is high, so it represents the share of the big pie going to each side. Write the seller's objective as g VH
d ,.., ~ = i=0
Po?o+Y
H
~-1
.
(8)
i=1
Since VH is given, the seller's problem is to choose the number of offers M, and the corresponding acceptance probabilities, so as to maximize the right-hand side of this expression. A crucial feature of the problem is the seller's inability to make commitments about future offers: after each rejection it must remain optimal to continue with the original offer sequence. Consider the seller's valuation when at least one offer has already been rejected, and k offers remain. Define k-1
Wk - 6k-~ [YO -- I ] p o + ~. [~ 6k-z -~ -- 3k-
1]Pi"
(9)
i=i
This is the contribution to the continuation value made by the last k offers, scaled by Va and discounted to the time when w k is offered. At this point, for k > 2, there is the alternative of making just k - 1 more offers, rather than k. That is, the seller could sweeten the current offer so as to advance the countdown by one period, inducing the buyer types who planned to accept next period to accept now instead, and so on. For k > 3 the condition needed to prevent the seller from switching to this alternative is Wk = [7 -- 6k- ']pk_ , + 8Wk_ , >_ [Y -- 5k- 2]pk_ , + Wk_ ,.
(10)
This is the basic optimality condition for the seller. It implies W k -- 1 ~-~ (~k -- 2 Pk - 1,
W k <-- 7Pk - 1"
(11)
Since the seller gains by shifting probability to earlier offers, the probability be reduced until condition (10) binds, implying that (11) holds as a pair
P k - 1 will
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of equalities, for k > 3. Thus 3k-2Wk-~-7Wk_l,
(12)
Wk--~7Pk_ 1"
For 2 < k < M - 1 these equations determine Wk and Pk recursively. In order to start the recursion, Pl must be determined: this is the smallest probability that will prevent the seller from pooling at the point where k = 2. The continuation value associated with pooling depends on the seller's assessment of the buyer's current type. Let (* be the screening threshold: this is defined as the solution of the equation (1 - 0 ( 7
- - 6) + 3 ( ( 7 o
-
1) =
0 + AU(~b( + 1 - tr) v.
(13)
This says that the expected value of pooling is the same as the value of making two offers, allowing for the fact that acceptance of the first offer reveals the high type, thereby increasing the seller's expected return from the next negotiation. The logic here is that 7,7o and the function U(o) are given, and these determine (*. Given the screening threshold, Pl is determined by the condition that the seller must be indifferent between pooling and screening when k = 2: ~* [pl + ~] = ~.
(14)
This gives W2 = [7 - 6]PI + 6[?o - 1]( = 609~, where the parameter co is defined by o~= [ ~ - I l Z *
+7o-1.
(15)
(16)
So
7k-2 Wk(?) -- 6 1/2~k- 3)k~0~.
(17)
Define the sequence nk as ~(7) = ~ -?T ~ -
1(?) -- ? k S - l/2k(~ + 3)
(18)
Then the optimal acceptance probabilities Pk are determined from the screening threshold (* and the variables (?, ?o) summarizing the continuation values as I4<. ?
p~(~,~) = ,+____!~; ~ _ ~ ( ~ ) ~ { .
(19)
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4.2. Determinin9 the number of offers For a given value of M, any reallocation of the probabilities Pi in favor of earlier offers increases the expected payoff. On the other hand, if too much weight is put on early offers then the screen will collapse as soon as these offers have been rejected. At each step in the recursion the proposed value ofpk is checked to make sure that it fits within the unit interval, and the recursion stops when this condition fails. Define k-1
/~k= 1-- ~ p~.
(20)
j=O
This is the largest probability that could be put on an initial offer which will be followed by k subsequent offers. The optimal number of offers must be such that this probability is insufficient to warrant extending the sequence from M to M + 1 offers. That is [7 - 6u]Pu + 6Wu < [Y - 6u - 1]Pu + Wra
(21)
d~ u < Wu.
(22)
SO
This, in combination with the reverse inequality warranting extension from M - 1 to M offers, implies W~
+Pl +P2 +"" +PM-2 < 1 - P o <---~ ( . / + Pl
d
M
M
k=3
k=3
+ P 2 + "'" +Pu-x,
Y YPl+ ~ Wk <--YEI--~]
(23)
which can be rearranged as " --~3 M y k - 3 1 1 - ~ ~ = 3 " k - 3Y (DOg (~l/2(k-2)(k-1)~ ( ( , ~ "~ (D(~R= t~l/E(k-~(k-1)"
(24)
Define 1
Z-~-
1,
1
Z*---~,
1.
(25)
Then M is determined by F l u - 3(7) <-
Z -Z* 09
< l-Iu- 2(Y),
(26)
J. Kennan/Japan and the World Economy 7 (1995) 447-472
458
where M
n~(7) = E ~s.
(27)
s=O
That is, there is a system of weights ~k(~,6) that determines the number of offers and the probabilities assigned to each offer, for any initial assessment ( that the state is currently low. The acceptance probability for the initial offer is the value implied by continuation of the recursion (12), plus the remainder of the unit interval (this remainder being insufficient to support an additional offer). That is,
PM-1 =PM-1
= 1-~-pl-
~ Pk = 1-k= 1
--00~17M_40' ) -~
--'°Ez7"
(28)
The seller's optimal continuation value relative to the continuation value of the high-type buyer, is U
~nn= [7--d]PM-1 + 6W~-1 =Ey-d]¢~o[Z~ Z* = (o~[(y-d) Z-Z• co
l'Iu_ 4(y)] + 6y~u_ 4(yKo~
-
7I'Iu- 4(7) + dFIu- 3(')]-
(29)
This ratio can alternatively be written as VU. = ¢o~[ (7 _d)Z-Z*+d_f~_3(7) o,
]
= ( 7 - - d ) [ 1 - ~ . ] + ~eo[d-f~t- z(Y)],
(30)
wheref~(7) is defined as
fu(7) = 71"Iu- 1 -
6M+ 2(I-Iu -- 1)
(31)
with the convention that II_ 1 = 0 so thatfo = O.
4.3. The buyer's continuation values Consider the value of the buyer after M - 1 screening offers have been rejected, and the seller has therefore concluded that the buyer is currently in the low state.
J. Kennan/Japan and the World Economy 7 (1995) 447 472
459
The value of the buyer at this point does not depend on which type of screen has just ended. Define VL as the value of the low-type buyer that has just signed a contract, and define Vn as what the high type would be worth now, if it had rejected enough offers to fool the seller. The incentive compatibility constraint implies that this is also the value the high type would have had now, if it had accepted some previous offer and thus revealed that the current profit rate is high. Thus, for example, the value of the high type at the start of negotiations is ~ M - 1 VH' and this does depend on the type of screen, because the number of offers will typically be greater in a hard screen than in a soft screen; also the low type's value at the start is 6 M- ~ VL, which again depends on the type of screen. The low-type buyer gets nothing from the current contract, so the value of the buyer, as of the signing date, is VL = AL(K ) + AXd~[p(K)VL + {1 -- p(K)} VH],
(32)
where d a = 6 n ° - 1 is the discount factor from the beginning to the end of a soft screen, and AL(K ) is the expected present value of the sequence of pooling contracts: K-1
AL(K ) = ~ A*[1 - p(s)].
(33)
s=l
In Eq. (32), VL and Vn are functions of K (among other things). If the seller never screens, then the valuation of the low-type buyer reduces t o AL(OO).Note also that if the contract is of infinite length, VL is zero, as in the basic screening model. If the high type imitates the low type, the yield from the current contract is vn - vL = 1, so the continuation value of the high-type buyer at the signing date is Vn = AH(K ) + AKda [p(K){1 - pa(K)} VL + a(K)VH-],
(34)
where K-1
AH(K ) = 1 +
K-1
~ ASt~(s)= ~ AS[z*+Cbsz *] s=l
1 -- A r
s=O
1 -
fix
= z~ 1 -----A+ z~ - 1- - - - ~
(35)
Here fl = A~ is a discounted persistence measure that will prove important in determining the buyer's cost of revealing that profits are currently high. These equations can be used to determine VL and Vn, for any given value of K and M a. The most convenient way to do this is to first solve for the difference Vn - VL. The transition rule of the Markov chain yields p(s) + a(s) - 1 = tk~.
(36)
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J. Kennan/Japan and the World Economy 7 (1995) 447-472
Therefore
_ilk ~-da/3k[V._ VL], v . - VL--11_/~
(37)
which implies 1 -/3 K
VH- K = 1-1 - 13] 1-1 -/3Kda]"
(38)
Finally, by substituting for An(K) and Vn - VL in the equation for VHabove, and using the relation 1 - a(K) = (1 - 4/)z*, the continuation values for the buyer in each state may be written in terms of model's basic parameters as
v.
1 - AK
.
1 -/3x
Zn(l* - a)(1 - Ard.) ~-ZL(1 -- fl)(1 -- flrd.)
(39)
and 1 - AK
1 - fir
VL = z~ (1 -- A)(1 - AXd.) - z~ (1 - fl)(1 - flXda)"
(40)
These results cover various interesting special cases. For example, if the buyer is now in the low state, its value depends on the difference between A and/3: if the contract is very long, or if there is a high degree of persistence, this buyer is not worth much, as one would expect.
Solution method In equilibrium the continuation values must reproduce themselves. This gives a system of three equations in the variables (y, (*, 09). The "(* equation", defining the screening threshold, can be written as 0 U° 609(* = ~ a + A-~n"
(41)
The ";~ equation" is ~= l+0+Ad
VH
b~'n + AUb
(42)
VH
The "?o equation" can be written as c°-(~-1)
Z * = 0 ( 1 -AK)/(1V. -A)+ArU°vH"
(43,
In each of these equations, the U/V piece is continuous in the variables (y, (*. co), but VHjumps when K changes, and also when M o changes (unless K is infinite); in
J. Kennan/Japan and the World Economy 7 (1995) 447-472
461
addition, the 7 equation jumps when M b changes. These j u m p discontinuities are the main source of difficulty in attempting to solve the model. The three equations to be solved can be expanded as
0 + A(7 -
6e)~* = ~
1 +O+Ad 7=
a o) 1 - ~
l + A~°(oEd ° _ f o ( ? ) ] ,
P.+A(7_d b 1
VH
+ A K~ao)Ida __fa (7)],
(44)
where ~o = e ~ * + 1 - tr,
~'~ = p ( K ) ,
(45)
~b = 1 - a,
a n d f a = f M , - 3, etc. Aside from the discontinuities, all three equations are linear in o), and the second two are linear in ~*, or, equivalently, in Z*. So if the second two are used to solve for to and ~* in terms ofT, and these are substituted in the first equation, the result is a single equation in ~. This will be a polynomial with j u m p discontinuities. The problem with this approach is that the values of the integer variables (Ma, Mb, Ms, K ) are not known until the system is solved. Pooling
If the opportunity cost of a conflict is high enough, or if the probability of discovering the high type is low enough, the seller always makes a pooling offer. In this case even when the seller is most optimistic, with ( = 1 - a, the return from pooling now must exceed the return from screening, given unconditional pooling in the continuation game. That is, -
(1-a)(7-6)-Sa(Vo
1-fl
1 ) ~ < 0 1 _- A p- ,
(46)
where 7 and 70 are determined from Eq. (7), with K = 0o and U b = 0/(1 - A). Then 1 - Ap
P'H -
VH -- (1 -- A)(1 -- fl)'
a - fl
(47)
(1 -- A)(1 - fl)'
SO (1 - A ) ( 1 - fl) + 0 ( 1 - fl) + A d ~ ( a - fl)
7=
1 -- Ap
'
0 ( 1 - - fl) ~o = 1 + 1 --A-----~"
(48)
462
J. Kennan/Japanand the World Economy 7 (1995) 447-472
If the seller will always pool in future then d b = 1, and the condition for optimal pooling now reduces to 1 -
Ap
1-fl
a
(49)
<~O.
1-a
If this holds with equality the seller is indifferent between pooling and screening when ( --- 1 - a. If the seller randomizes in this situation then 6 < d b < 1. This gives a region defined by (1
-
a A)l_a
1 -- Ap a - 1-fl l-a"
~< 0 ~< - -
(50)
Thus the equilibrium is completely determined whenever the first of these inequalities holds: the seller either screens randomly at (b or always pools. 5
The Coase property
The Coase property provides a useful reference point in interpreting the equilibrium, and it may also prove useful as a computational device. Let 6 approach 0 while A is held fixed, representing a situation where the seller's commitment power is negligible but the duration of each contract is fixed. In this case Eq. (8) implies that ui = Uo, for all i, so 7 = 70. Also z* A +
VH=I--
z*
1--fl
_
1 -
Ap
(l -- A)(1-- fl)
(51)
and 0 U= Ua=u o = 1-A
(52)
so that the seller's valuation is just the pooling value. Approximate equilibria
The Coase property can be used to find an approximate solution of the model. Suppose that the Coase property will apply in the next contract negotiation, but the seller has some commitment power in the current negotiation. In this case the screening threshold for the current contract is the value of ( that solves the equation (1 -
0(V*
-
6) + 6ff(7" -
1) = 7 " -
1,
(53)
5When K = oo and Mb< 3 the ratchet effectdoes not cause trouble: the ratchet condition (described below) holds for all ft.
J. Kennan/Japan and the World Economy 7 (1995) 447-472
463
which implies ~*
1
1 - Ap
7*
1-Ap+O(1-/~)'
7*(7* - 1)
to=
(54)
6
These parameter values can be used to generate an approximate equilibrium. Transient information When there is no persistence in the private information process, the model simplifies considerably, although the repeated contract structure still causes complications relative to the static case. The seller's continuation value does not depend on the outcome of the previous contract, so the right sides of the ( and 7o equations are identical. Subtracting one of these equations from the other gives a solution for to in terms of 7 and ~*; then subtracting the 7 equation from the equation and substituting for to yields 1
7(7-
~* = - , ~,
co = - c5
1)
(55)
The V equation can then be expressed as a monotonic function of ~, with jumps where M changes, so existence of a unique solution is guaranteed. This model is analyzed in detail in Kennan (1993). Numerical solutions The following ad hoc procedure was used to solve numerical examples. First, the problem can be reduced to two equations in two unknowns by using the 70 equation to solve for to as a function of(* and y. This equation is piecewise linear in to and since K is determined by (*, and U/VH is continuous in to, jumps discontinuities with respect to occur only where M, changes. Write the equation as
v.
+
-1
Z*+A~(~-d")
1-~
-R, to=0,
(56)
where R a ~ 1 - mK~a[da - f ~ ( 7 ) ] "
(57)
This is piecewise linear in to, and continuous except for the first term, which jumps when d a changes. The continuous part is decreasing in 09, since R a is positive. Also, VH is increasing in d a, so the first term jumps down when d" increases. As to increases the test ratio that determines M decreases, so when to passes a critical value M, decreases, da increases, and the first term jumps down. The upshot of all this is that the function is everywhere decreasing, with downward jumps where
J. Kennan/Japan and the Worm Economy 7 (1995) 447-472
464
~o crosses a threshold. If the function jumps across the axis then randomization over two adjacent values of M a is needed to get a solution. To compute the solution start by assuming Ma = 2, which means Z ° - Z* < o~. When the equation is evaluated at this threshold, the result is
O(1-Ar)/(1-A) v.
'Z,_Za
~~
[
(a]
+ AK~ 1 - ~
.
(58)
If this expression is positive, then the root must lie beyond this threshold, meaning M~ = 2. A negative value is possible because Z" exceeds Z*. In this case test the equation at the next threshold, which is Z ~ - Z * = e)II1(,)= ~° [ 1 + ~ 2 ]
(59)
meaning M. = 3, and continue until a positive value is found. This pins down M., and reduces the problem to solving a linear equation in o~. The solution is substituted in the ( and 7 equations, leaving two equations in the two unknowns
(~, (*). The next step is use the ( equation to solve for 7 as a function of (*. With (* given, each value of 7 generates values for all of the integer variables. Then the equation can be evaluated as a function ofT, and solved numerically. Finally, the result is substituted in the 7 equation, which is then solved numerically for (*. In order to handle the discontinuities, the numerical solutions are obtained using a homemade algorithm that starts with an interval where the equation changes sign, splits this interval in half at each iteration, and solves a linear approximation joining the end-points. Thus the length of the interval shrinks by at least a factor of 2" after n iterations. If the algorithm continues for a large number of iterations without finding a solution, the conclusion is that the equation jumps across the horizontal axis, so randomization is needed. An illustration of the computational difficulties is given in Fig. 2, which plots the (* and ~ equations as functions of Z* after substituting for ~o. The continuous parts of both equations are downward sloping (for the parameter values used in this example), and there is more action in the ~, equation, which jumps one way when Ma changes, and the other way when Mbchanges. Fig. 3 shows the final step of the solution process, in which the 7 equation is regarded as a function of Z*, after substituting a value of 7 that (numerically) solves the ~ equation at each value of Z*.
The ratcheteffect Screening is feasible in this model only if the ratchet effect is not too strong. The penultimate offer must be sufficiently attractive to induce the high-type buyer to forgo a sequence of K potentially lucrative pooling offers, submitting instead to
J. Kennan/Japan and the World Economy 7 (1995) 447-472
465
\
1.15-
\ 1.145gamma
1,14-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
zeta"
Fig. 2. y and (* equations. a hard screen in the next contract negotiation. But the amount needed to attract the high type might also be more than enough to attract the low type, in which case the screening equilibrium breaks down. For instance, Hart and Tirole (1988, Proposition 3) showed that in the fixed-valuation case (tk = 1) with single-period 1 (rental) contracts and ~ > ~, the union makes only pooling offers in equilibrium (unless the horizon is finite, and close at hand). More generally, the feasibility of screening depends on whether the discounted persistence parameter fl is sufficiently small.
466
J. Kennan/Japan and the World Economy 7 (1995) 447-472
J
0.001!
0.00'
L-
f 0.3
0.4
0.5
0.6
0.7
0.8
Z
-o.ooo~
Fig. 3. Reduced y equation as a function of Z.
Let V° denote the value of the low-type buyer accepting wl, given that the union incorrectly infers that the buyer is a high type. Then V ° = 0 - w 1 + Adb[(p Vt + {(1 - p}) Vn)].
(60)
Comparing this with Eq. (3) yields
~(EV. --
o ) = 1 "-F f ~ A d b E ( V H - VL] ). VL]
(61)
The low type's incentive compatibility condition can be stated as ~(VH - V°H)/> 6(VH - VL) and then, using Eq. (38), the feasibility condition for an
J. Kennan/Japan and the World Economy 7 (1995) 447-472
467
equilibrium with screening becomes
[6 - fldb] [1 -- fir] << [1 -- fl] [1 - daflx].
(62)
This can be written as 0 ~< hK(//) = 1 -- ~ --//(1 -- rib) +/~x(6 _ da) +//K+ l(d ~ _ rib).
(63)
This condition is difficult to check, since K, d, and db are not k n o w n in advance. A simple case arises when either there is no persistence across contracts or else the contract is infinitely long, so t h a t / / = 0: then the condition reduces to 6 _< 1, and the screening equilibrium is feasible. T h e fixed-valuation case is also transparent: h e r e / / = A, the union always m a k e s pooling offers after a screen reveals the low type, and always d e m a n d s v . after the high type is revealed, so K = oo, db = 0, and the condition reduces to ~ _ 1 - A, or 6 _< ½ for rental contracts (where T = 1 so ,~ = A). M o r e generally, the acceptable range f o r / / c a n be analyzed by first considering the case where K = 1, and then c o m p a r i n g hK(//) with hi(~~ ). F o r any K, hK(0 ) = 1 - 3 and h ~ ( 1 ) = 0 , while hi(~~) is a quadratic with roots at 1 and (1 - 5)/(do - db). Also,
hK + ~(//) -- h/c(//) = / ~ x ( / / _ 1)[6 - d,, + ~(a, - db)]
(64)
w i t h / / < 1 and 6 _> d, _> d b, so the function hK+ 1(//) lies below hK(//). The situation is illustrated in Fig. 4 for Ma = 2 and M b = 5. The admissible values o f / / a r e those below the smaller root of hK(//). If K is infinite, the ratchet condition is satisfied
The Ratchet Condition M..s-2, M_b-5
0.11114
\ 0.0~ 0.000
..: ? ~ . •,~... ~
.-.____
" . "...., -0.0~
"4
". . . .
-0.004 0.0 0.1 0.2 0.8 0.4 0.5 0.6 0.7 0.8 0.9 1.0 8 --K-1
"Km2--K-5
F i g . 4.
J. Kennan/Japan and the World Economy 7 (1995) 447-472
468
when fl is below (1 - 6)/(1 - db). Reducing K allows fl to increase slightly, but the upper bound of(1 - 6 ) / ( d a - d b ) means that fl must be small unless strikes are very short. Thus there may be a tradeoff between allowing the model to explain long strikes, implying virtually no persistence across contracts, or allowing it explain linkage across contracts, with strikes that are too short to match the data.
5. Examples Some examples of the above results are shown in Table 2. The examples show that equilibria of the model can be found that match the intuitive description given at the outset. The seller goes through regular cycles of optimism and pessimism, with unsuccessful screens being followed by a depressed pooling phase, ending with a very short screen, while successful screens generate optimism about the gains from screening in the next negotiation. The length of the pooling phase, and the maximal duration when the union screens, respond to changes in the transition probabilities, and to changes in contract length. The examples suggest that if there is substantial persistence, the contract length must be very long in order to avoid the ratchet effect. The discount factor 6 is Table 2 Numerical examples
vL vH wo 6 p a T d~ z* A 7 ~* ~o w1 K VL
Vn Ma Ua Mb Ub
h(fl)
A
B
C
D
E
F
1/11 12/11 0 0.9975 0.98 0.9 1200 0.88 0.833 0.0496 1.0906 0.8844 0.10356 0.091 10 0,0011 1.0467 2 0.0957 23 0.1304 0.0002
1/I 1 12/11 0 0.9975 0.95 0.9 1200 0.85 0.667 0.0496 1.0906 0.8829 0.1038 0.091 3 0.0027 1.0467 2 0.0957 23 0.1303 0.0002
1/11 12/11 0 0.9975 0.95 0.6 1200 0.55 0.889 0.0496 1.0925 0.9047 0.1028 0.0928 4 0.0027 1.0307 2 0.0957 10 0.1035 0.002
1/11 12/I 1 0 0.9975 0.95 0.6 900 0.55 0.889 0.0496 1.0946 0.8841 0.1068 0.0927 ~ 0.0062 1.0676 1 0.0952 10 0.1021 0.0012
1/11 12/11 0 0.9975 0.1 0.9 150 0 0.1 0.687 1.1537 0.8667 0.1778 0.0978 1 1.76 2.76 16 0.4583 16 0.4583 0.0025
0.0934 1.0934 0 0.998 0.1 0.9 150 0 0.1 0.687 1.156 0.865 0.18 0.1
1 1.767 2.767 15.45 0.491 15.45 0.491 0.0025
469
J. Kennan/Japanand the World Economy 7 (1995) 447-472
intended to refer to offers spaced one week apart, as in Kennan and Wilson (1991), so a 3-year contract corresponds roughly to T = 150. Thus the contract length in column E is realistic, but this example assumes no persistence, so that the equilibrium involves screening in each negotiation. Examples A - C show that sequences of pooling offers are made when the degree of persistence is relatively high, but the contracts in these examples last about 24 years.
5.1. Randomization by the uninformed seller
Randomization by the informed buyer is a familiar feature of static screening models in which the distribution of buyer types is discrete. A novelty of the repeated screening model is that randomization by the uninformed seller may also be required in order to reach an equilibrium. During the countdown, the
1,I~5
1.1~
1.1~5
Gamma 1.12
1,1565
Fig. 5. Randomization.
1,157
1.15~
1.1~
470
J. Kennan/Japan and the World Economy 7 (1995) 447-472
seller is indifferent at each stage between making k or k - 1 more offers and it is important that this tie is broken in favor ofk. But if the parameter values are just right, the seller is indifferent at the outset between M and M + 1 offers, and in this case any random mixture of these choices will do equally well, from the seller's point of view. Equilibrium requires that the seller chooses the right mixture in this situation. An example of this is in column F of Table 2. Fig. 5 shows the ? equation for this example, as described in Section 4, with the (unique) solution lying on the vertical section of the curve.
5.2. Experienced bargainers do not fight It is generally found that conflicts in labor negotiations are most severe when the union is negotiating its first contract. Column D in Table 2 contains an example in which there is no private information in the long run, because the probability of the high state, according to the stationary distribution of the Markov chain, is too low, relative to the cost of conflict, to warrant screening. Nevertheless, if the union is initially optimistic for some reason there will be a screen in the first contract, and if the high state is discovered there will again be a screen in the next contract, and so on until the low state is discovered after which the union makes pooling offers in all subsequent contracts.
Acknowledgments This paper was presented at the Ninth Annual Japan-US Technical Symposium, New York University, March 1994. The paper grew out of a joint research project with Robert Wilson, to whom the author is grateful for many enlightening discussions. The National Science Foundation provided the research support.
References Blume, A., 1990, Bargaining with randomly changing valuations, Working Paper 90-22, University of Iowa. Borland, J. and J. Tracy, 1988, The determinants of expected contract duration and early renegotiation: An empirical analysis, Yale University, unpublished. Card, D., 1987, A longitudinal analysis of strikes and wages [-draft of Card (1990b)], Princeton University, April. Card, D., 1988, Longitudinal analysis of strike activity, Journal of Labor Economics 6, 147-176. Card, D., 1990a, Strikes and bargaining: A survey of the recent empirical literature, American Economic Review 80(2), 410-415. Card, D., 1990b, Strikes and wages: A test of an asymmetric information model, Quarterly Journal of Economics 105, 625-659. Cramton, P.C., 1984, Bargaining with incomplete information: An infinite horizon model with two-sided uncertainty, Review of Economic Studies LI(4), (167), 579-594.
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Cramton, P. C., 1992, Strategic delay in bargaining with two-sided uncertainty, Review of Economic Studies 59, 205-225. Cramton, P. C. and J. S. Tracy, 1992, Strikes and holdouts in wage bargaining: Theory and data, American Economic Review 100-121. Cramton, P. C. and J. S. Tracy, 1994a, The determinants of U.S. labor disputes, Journal of Labor Economics 12(2), 180-209. Cramton, P. C. and J. S. Tracy, 1994b, Wage bargaining with time-varying threats, Journal of Labor Economics 12, forthcoming. Fischer, S., 1977, Long-term contracts, rational expectations and the optimal money supply rule, Journal of Political Economy 85, 191- 205. Fudenberg, D., D. Levine and P. Ruud, 1985, Strike activity and wage settlements~ UCLA Working Paper 249, revised September 1985. Fudenberg, D. and J. Tirole, 1983, Sequential bargaining with incomplete information about preferences, Review of Economic Studies 50, 221 247. Gray, JoAnna, 1978, On indexation and contract length, Journal of Political Economy 86, 1 18. Harris, M. and B. Holmstrom, 1987, On the duration of agreements, International Economic Review 28, 389-406. Harrison, A., 1993, Strike and contract data for postwar canada, McMaster University, (unpublished draft.) Harrison, A. and M. Stewart, 1989, Cyclical variation in strike-settlement probabilities, American Economic Review 79, 827-841. Hart, O., 1989, Bargaining and strikes, Quarterly Journal of Economics 104, 25-44. Hart, O. and J. Tirole, 1988, Contract renegotiation and coasian dynamics, Review of Economic Studies 55, 509-540. Hayes, B., 1984, Unions and strikes with asymmetric information, Journal of Labor Economics 2, 57-83. Hicks, J. R., 1932, The theory of wages (Macmillan, New York). Ingram, P., D. Metcalf and J. Wadsworth, 1991, Strike incidence and duration in british manufacturing industry in the 1980s, Working Paper No. 88, Centre for Economic Performance, LSE, April 1991. Joskow, P. L., 1987, Contract duration and relationship-specific investments: Empirical evidence from coal markets, American Economic Review 77, 168-185. Kennan, J., 1986, The economics of strikes, in: O. Ashenfelter and R. Layard, eds., Handbook of Labor Economics, Vol. II (Elsevier, Amsterdam). Kennan, J., 1993a, Repeated contract negotiations with transient private information, presented at the Society for Economic Dynamics and Control, Annual Meetings, Nafplio, June 1993. Kennan, J., 1993b, Information cycles in repeated contracts, presented at the NBER Summer Institute, July 1993. Kennan, J. and R. Wilson, 1989, Strategic bargaining models and interpretation of strike data, Journal of Applied Econometrics 4, $87-S 130. Kennan, J. and R. Wilson, 1990a, Can strategic bargaining models explain collective bargaining data? American Economic Association Papers and Proceedings, 80(2), 405-409. Kennan, J. and R. Wilson, 1990b, Theories of bargaining delays, Science 249, 1124-1128. Kennan, J. and R. Wilson, 1993, Bargaining with private information, Journal of Economic Literature 45-104. Kennan, J. and R. Wilson, 1991, Screening models of bargaining with private information: An empirical application, unpublished draft. McConnell, S., 1989, Strikes, wages, and private information, American Economic Review 79, 801-815. Riddell, W. C., t979, The empirical foundations of the phillips curve: Evidence from Canadian wage contract data, Econometrica 47, 1-24.
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Riddell, W. C., 1980, The effects of strikes and strike length on negotiated wage settlements, Relations Industrielles 35(1), 115-120. Sobei, J., 1989, Durable goods monopoly with entry of new consumers, UC San Diego, (unpublished.) Sobel, J. and I. Takahashi, 1983, A multi-stage model of bargaining, Review of Economic Studies 50, 411-26. Tracy, J. S., 1986, An investigation into the determinants of U.S. strike activity, American Economic Review 76, 423-436. Vincent, D., 1990, Bilateral monopoly, non-durable goods and dynamic trading relationships, Northwestern University, CMEMS #832, revised January 1990. Vroman, S., 1989, A longitudinal analysis of strike activity in U.S. manufacturing: 1957-1984, American Economic Review 79, 816-826.
Japan and the World Economy 7 (1995) 447-472
/" Z a n and the WORLD ECONOMY
Repeated contract negotiations with private information John Kennan* Department of Economics. University of Wisconsin, Social Science Building, 1180 Observatory Drive, Madison, WI 53706, USA
Received April 1994; accepted December 1994
Abstract Labor contracts are repeated, and the current negotiation apparently affects negotiations on the next contract. There is evidence that the likelihood of a strike is influenced by what happened before the previous contract was signed. To analyze whether private information might explain this, a model of negotiations between a buyer and a seller is developed where the "pie" in each contract follows a Markov chain, with transitions observed privately by the buyer. Each negotiation is a sequence of offers by the seller; the buyer can only accept or reject each offer, and a contract is signed when the buyer accepts. Keywords: Bargaining; Private information; Labor contracts; Strikes JEL classification: C78; D82; J41; J51
1. Introduction M a n y prices are negotiated in the context of repeated bargaining relationships, with an element of bilateral m o n o p o l y involved. The leading example is the price of labor services, set in union contracts that are renegotiated every few years; other examples include contracts for coal, steel and other raw or intermediate inputs. Uncertainty about the other party's reservation price is apparently a standard feature of these negotiations. * Tel.: 608-262-5393. Fax: 608-263-3876 or 608-262-2033. E-mail:jkennan(b~macc.wisc.edu. 0922-1425/95/$09.50 © 1995 -Elsevier Science B.V. All rights reserved SSD! 0922-1425(95)00020-8
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J. Kennan/Japan and the World Economy 7 (1995) 447-472
Recent work on labor contracts has emphasized the possibility of explaining collective bargaining outcomes in terms of the incentive-compatibility or truthtelling constraints arising when either party to a bargaining game is endowed with unverifiable private information. 1 The central point in this work is that outcomes such as strikes that appear inefficient can be seen as a necessary cost of extracting information. Similarly, theoretical analyses of macroeconomic fluctuations have emphasized the possibility that labor contracts might prescribe layoffs as an efficient response t ° an employer's privately observed signals about the value of labor's product, even though the laid off workers' marginal product exceeds their reservation wage. The empirical literature on labor contracts has shown convincingly that the outcome of the current negotiation is substantially influenced by what happened when the previous contract was negotiated. 2 This paper develops a model of repeated negotiations with private information on the buyer's side that can be used to confront some of the regularities found in the labor contract data. The buyer's valuation changes according to a two-state Markov chain, and in equilibrium the seller goes through cycles of optimism and pessimism, triggered by whether aggressive bargaining tactics are successful in each negotiation. In the labor context the union calls a strike periodically in order to update its information about the employer's valuation, and a long strike with a low wage settlement at the end induces the union to lie low in the next few contract negotiations. In order to obtain concrete results it is necessary to adopt a specific set of bargaining rules, and a stochastic process for the buyer's valuation. Here it is assumed that the uninformed seller makes offers that the buyer must either accept or reject, so the equilibrium involves screening or price discrimination. A twostate Markov chain is the simplest interesting process that allows the buyer's valuation to have both transient and permanent components. At one extreme, no transitions occur so that the high-valuation buyer is very wary of revealing its type, because of the "ratchet effect": once the high valuation is revealed, the seller will claim the entire rent in all future negotiations. At the other extreme, the current valuation is entirely transitory, so that the model reduces to a repeated sequence of one-shot screening negotiations. Between these extremes the model is capable of generating equilibria in which the seller makes either a pooling offer or a sequence of screening offers in any given contract negotiation, depending in part on what happened in the previous negotiation. It is assumed throughout this paper that contract duration is given, and that the Markov chain makes one transition for each contract. The literature contains no good theory of contract duration: we do not know why unions 1 See, for example, Hayes (1984), Fudenberg et al. (1985), Kennan (1986), Hart and Tirole (1988), Hart (1989), Kennan and Wilson (1989, 1990, 1993), Card (1990a, b), and Cramton and Tracy (1992, 1994a, b). 2 See, for example, Riddell (1979, 1980), Card (1988, 1990b), and Ingram et al. (1991).
J. Kennan/Japan and the World Economy 7 (1995) 447 472
449
and employers sign contracts in the first place, and we do not know why these contracts often last three years, sometimes one, two or four years, but rarely longer) There is a possibility that the model developed here might shed light on this question, by linking the contract duration to the degree of persistence in the buyer's private information. In order to explore this one must first analyze the equilibrium for a given contract duration and a given degree of persistence, and then consider how the equilibrium changes as these parameters are varied. The model developed here is related to earlier work by Blume (1990) and Vincent (1990) on the effects of new information that arrives while a bargaining game is in progress. Blume (1990) analyzed a two-type model where the low type can temporarily assume the valuation of the high type, and emphasized that even if the informed party is limited to accepting or rejecting offers made by the uninformed party, there is an important signaling aspect of the negotiations. The model in this paper differs from that of Blume's in two respects: the information structure is a two-state Markov chain, so that both types might change valuations, and the game involves repeated contract negotiations, as opposed to a final sale.
2. Multiple negotiations A serious limitation of the recent literature applying strategic bargaining models to labor negotiations is that each contract is treated in isolation. In practice, labor negotiations involve contracts covering several years, and the parties seem mindful of how the outcome of the current contract will affect negotiations on the next contract. A natural theoretical interpretation is that private information is to some extent temporary, but also to some extent permanent, so that if either side lets the cat out of the bag in the current negotiation, its bargaining position will be weaker when the next contract is negotiated. Recent evidence developed by Card (1988, 1990b) indicates that the likelihood of a strike is heavily influenced by whether a strike occurred before the previous contract was signed, and, if so, by the duration of the strike (see Table 1). Similarly, Ingram, et al. (1991) found that the likelihood of a strike in British wage negotiations is strongly influenced by what happened in the previous year's negotiations: curiously, a "go slow" last year means that a strike is less likely this year, while an overtime ban last year has the opposite effect. At first sight it seems odd that short and long strikes last time would have opposite effects on the strike probability this time. But a private-information 3See, for example, Fischer (1977),Gray (1978), Harris and Holmstrom (1987),Joskow (1987)and Borland and Tracy (1988).
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J. Kennan/Japan and the World Economy 7 (1995) 447-472
Table 1 Outcomes of successive contract negotiations Canada
US Current strike duration
Currentstrike duration
Previous strike duration
Total
None
Short
Long
Total
None
Short
Long
None
2198 82
Long
235
Total
2515
67 3.0% 7 8.5% 5 2.1% 79 3.1%
174 7.9% 25 30.5% 29 12.3% 228 9.1%
3651
Short
1957 89.0% 50 61.0% 201 85.5% 2208 87.8%
3105 85.1% 136 61.3% 411 78.7% 3652 83.1%
179 4.9% 35 15.8% 26 5.0% 240 5.5%
367 10.1% 51 23.0% 85 16.3% 503 11.4%
222 522 4395
Notes: The US results are derived from the data used by Vroman (1989); similar results appeared in Card (1988). The results for Canada are derived from a dataset assembled by Harrison (1993), using contract and strike data files from Labor Canada. A short strike means that the duration was 1-14 days (1-10 working days in the Canadian Data).
m o d e l m i g h t e x p l a i n this n o n m o n o t o n i c effect as follows. A s h o r t strike last time m i g h t m e a n t h a t the firm was f o u n d to be rich, a n d is p r o b a b l y still rich, so the u n i o n m a k e s an aggressive d e m a n d which will cause a strike if the firm's c u r r e n t profits d o n o t meet expectations. A peaceful settlement last time m i g h t m e a n t h a t the u n i o n was n o t sufficiently o p t i m i s t i c to t h r e a t e n a strike, a n d thus no new i n f o r m a t i o n was revealed. F i n a l l y , a l o n g strike last time c o u l d m e a n t h a t the firm was f o u n d to be p o o r , a n d so the u n i o n d o e s n o t c o n s i d e r it w o r t h w h i l e to t h r e a t e n a strike now.
3. An Infinite horizon Markov model of repeated negotiations C o n s i d e r a n infinite sequence of c o n t r a c t n e g o t i a t i o n s b e t w e e n a b u y e r a n d a seller where the rent to be d i v i d e d in each T - p e r i o d c o n t r a c t follows a simple M a r k o v process, with t r a n s i t i o n s t h a t are o b s e r v e d p r i v a t e l y by the buyer. B o t h sides m a x i m i z e the p r e s e n t value of expected income, with a c o m m o n d i s c o u n t factor 3 p e r period. F o r e x a m p l e , if v is the present value of a firm's r e v e n u e d u r i n g the t e r m of a l a b o r contract, net of all n o n l a b o r costs, a n d if Wo is the highest w a g e a v a i l a b l e to w o r k e r s d u r i n g a strike, then the rent is v - w o. A s s u m e t h a t the rent follows a t w o - s t a t e M a r k o v c h a i n with c o n t i n u a t i o n p r o b a b i l i t i e s p a n d a over the length of a c o n t r a c t , so t h a t if the rent is low n o w (v = VL), it will a g a i n be low T p e r i o d s hence with p r o b a b i l i t y p, a n d if the rent is high n o w (v = vR) it will be
J. Kennan/Japan and the World Economy 7 (1995) 447-472
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high again next time with probability a. 4 It is convenient to use w 0 as the origin and vH - v L as the unit, so relabel 0 = (vL -Wo)/(v n - V L ) as the low rent, with 1 + 0 as the high rent and zero as the seller's opportunity cost. Let A = fix be the discount factor from the beginning to the end of a contract. Then the model is summarized by the five parameters (p, a, 0,6,A). For empirical purposes, A m a y be treated as approximately observable, in the sense that it must correspond to some market discount factor. On the other hand 6 is the discount factor covering one bargaining period, defined as the length of time between offers in each contract negotiation, and this will be treated as unobservable. In the absence of any historical information, the probability of the low valuation is that implied by the stationary distribution of the M a r k o v chain, i.e. 1 -a
Z*=l--tx+l-p
1 -a
1--~'
(1)
where ~b = p + a - 1 measures the degree of persistence in the M a r k o v chain. Let p(s) denote the probability that the M a r k o v chain is in the low state after s transitions, given that the current state is low, and let a(s) be the corresponding probability for the high state. Then p(s) = z* + eSz*,
,~(s) = z~, + 4~Sz~.
(2)
The parameter @governs the extent to which successive contract negotiations are linked. Assume ~b/> O, so that the probabilities p(s) and a(s) do not oscillate. If (~ = 0 information is completely transitory, so that any inference that the seller might draw from the current contract negotitation will be irrelevant by the time the next contract is negotiated. At the other extreme, if ~b = 1 the current information is entirely permanent. Under the interpretation that @ summarizes multiple transitions during the term of a contract, with ~b = @T, the linkage across contracts is made weaker if the contract length is increased, for a given value of ~bo. In the most interesting parameter configuration, the seller would not find it optimal to make a pooling offer unless there is recent information indicating that the buyer is likely to be p o o r this time. On the other hand, if the seller inferred from the previous outcome that the rent was low, it may be optimal to make a pooling offer in the current negotiation (i.e. to make an offer which all types of the buyer will surely accept). This means that the seller learns nothing new in the current negotiation, so when the next contract comes up, the seller's information will be weaker than it is now. If the information process is highly persistent, 4 This notation may be taken as a summary of multiple transitions in the Markov chain during the term of the contract: for instance if the chain makes one transition per period, and the basic transition matrix is Ao, then the transition matrix from one contract to the next is A = Aov, and p and a may be interpreted as the diagonal elements of this matrix. Under this interpretation vL and vn denote expected present values over the life of the contract, given the state in the initial period.
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J. Kennan/dapan and the World Economy 7 (1995) 447-472
pooling may still be optimal next time, but eventually the seller's information erodes to the point where it again becomes optimal to screen. This provides an alternative interpretation of the following passage from Hicks (1932): "Weapons grow rusty if unused... The most able Trade Union leadership will embark on strikes occasionally, not so much to secure greater gains upon that occasion.., but in order to keep their weapon burnished for future use..." (p. 146).
In the screening model it is the union's information rather than its strike weapon which grows rusty, but the effect is similar. A Markov-Perfect equilibrium of this game is a renewal process based on the outcome of screening offers made by the seller. Thus if the buyer accepts an offer revealing that the rent is currently high, the continuation game is the same as it was the last time such a revelation was made, and similarly if the buyer rejects sufficiently many offers to convince the seller that the rent is currently low. In each contract negotiation there are two possibilities from the seller's point of view. If information is sufficiently persistent (d~ is relatively high) and if the seller has inferred from a recent negotiation that the rent was low, it will be optimal to make a pooling offer. Alternatively, if the seller believes that the high-rent state is sufficiently likely, a sequence of screening offers will be worthwhile; these offers will be acceptable to the buyer if the rent is currently high, and unacceptable if the rent is low. If the buyer accepts a screening offer, the seller will infer that the rent is high, and so the seller will screen again when the next contract is negotiated (unless perpetual pooling is optimal). Of course the buyer knows that acceptance of a screening offer weakens its bargaining position next time, so the offer must be sufficiently generous to compensate for this. If the buyer rejects all screening offers, on the other hand, the seller infers that the rent is currently low, and it may then be optimal to make a pooling offer next time, and perhaps again the time after that, and so on. A key feature of the equilibrium is the number of pooling offers, K - 1, made by the seller in the sequence of contracts following rejection of a screening offer. The equilibrium is derived by considering the continuation values of the game for the buyer and for the seller, in the various circumstances that may prevail when the seller screens. Screening negotiations last through M offers, and then a T-period contract is signed; let vH, vL and w represent present values of T-period streams as of the signing date. After rejection of M - 1 offers the seller concludes that the rent is currently low, and the seller then makes pooling offers in the next K - 1 negotiations, followed by a screen in the Kth negotiation: call this a "soft" screen, and let U a be the seller's continuation value in this case. If the buyer accepts an offer revealing a high rent now, the seller screens again when the next contract comes up: call this a "hard" screen, with continuation value U b for the seller. This is sketched in Fig. 1, which represents varying degrees of pessimism for the seller. At one extreme, the seller believes the buyer is in the low state now
J. Kennan/Japan and the World Economy 7 (1995) 447-472 Hard
Soft
Screen
Screem
453
Pool
Screen
¢, m
1-¢~
z~
o(K)
~*
P
Fig. 1. The screening and pooling cycle.
with probability p, because there was a screen in the previous negotiation and the buyer was found to be poor. In this situation the seller pools now, and pools again in K - 1 successive negotiations until the probability of the low type has decayed past the screening threshold (*. At this point there is a screen, but the screen is "soft" in the sense that the seller is still relatively pessimistic: the probability of the low state is p(K), which is above the stationary value z~, and so the buyer will not have to reject many offers before the seller gives in. At the other extreme, the seller is most optimistic after the buyer was found to be rich in the previous negotiation; then the probability of the low state now is only 1 - a, so the seller's posture is "hard" in the sense that it will persist through relatively many rejected offers before giving in with a pooling offer.
4. Equilibrium An equilibrium in this model is a reproducible set of continuation values for the buyer and the seller at the point of signing a contract. That is, if both sides take these continuation values as given for all contracts after the current one, and bargain optimally now, then the continuation values at the end of the current negotiation must reproduce the equilibrium continuation values. The equilibrium will be constructed in three steps. First, the sequence of offers that would be acceptable to the high-type buyer is derived from the condition that each of these offers must give the high-type seller the same continuation value as would be obtained by refusing all but the last offer, and causing the seller to infer (wrongly) that the current state is low. Secondly, the seller must determine how much weight to put on each of these offers, subject to the constraint that sequence of remaining offers must be credible after each offer is
J. Kennan/Japanand the Worm Economy 7 (1995) 447-472
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rejected. Finally, the state-contingent continuation values thus calculated must reproduce themselves. In a screening negotiation the seller makes a sequence of offers such that the high-type buyer is indifferent between accepting each offer and waiting for the next one. In equilibrium the high type randomizes so that after M - 1 offers have been made, all high types have accepted, and the seller then offers 0, which is accepted by the low-type buyer. Define VH as the continuation value of the game for a high-type buyer that has just signed a contract at 0, by imitating the low type. During the countdown from the seller's initial offer, labeled wM_ 1' to the last offer wo = 0, the buyer must be indifferent between the last offer and the offer wi that is made when there are i periods left in the countdown. Thus
6iVn=l+O-wi+Adb~'n,
i>l,
Vn - [1 - tr] VL +
(1 -
where d b = screen.
6ub- 1
a V H = V n --
p ) ( V n --
VL),
(3)
is the discount factor from the beginning to the end of a hard
4.1. O p t i m a l s c r e e n i n g
Define U a and U ~ as the seller's continuation values at the start of future soft and hard screening negotiations, which are taken as given when the seller constructs an optimal sequence of offers for the current contract. Let u~ be the seller's continuation value if the buyer accepts w~ in the current negotiation. Then
ui=-wi+AUb=I+O+Adb~'H-OIVH+AU b, i>_l
(4)
and 1- A r u o = O~_A+
AKua"
(5)
Suppose that at the start of the current negotiation the probability that the buyer's valuation is low is (, according to the seller's information. Then if Pi is the probability that the buyer accepts w~, with Po = (, the seller's valuation is M - 1
M-
U= 2 tSM-l-ipltti=d ~ i=0
1 1J~i''i
t$i'
(6)
i=0
where d = 6 M- 1 is the discount factor from the beginning to the end of the sequence of offers. Since the contingent valuations u i depend only on how the
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455
game continues after the current negotiation ends, the seller must take them as given, and maximize by choosing the probability sequence {Pi}. Everything the seller needs to know about future contracts is summarized by the screening threshold (* and the pair (7, 70) defined by 1 + 0 + A U b + Adb~"n
VH
~' -
Yo- 1 +
ui
- VH t- '~'
0(1-AX)/(1-A)+AKU "
v.
uo = - - + 1.
v.
(7)
Here Vis the ratio of the joint continuation value to the buyer's value, given that the current state is high, so it represents the share of the big pie going to each side. Write the seller's objective as g VH
d ,.., ~ = i=0
Po?o+Y
H
~-1
.
(8)
i=1
Since VH is given, the seller's problem is to choose the number of offers M, and the corresponding acceptance probabilities, so as to maximize the right-hand side of this expression. A crucial feature of the problem is the seller's inability to make commitments about future offers: after each rejection it must remain optimal to continue with the original offer sequence. Consider the seller's valuation when at least one offer has already been rejected, and k offers remain. Define k-1
Wk - 6k-~ [YO -- I ] p o + ~. [~ 6k-z -~ -- 3k-
1]Pi"
(9)
i=i
This is the contribution to the continuation value made by the last k offers, scaled by Va and discounted to the time when w k is offered. At this point, for k > 2, there is the alternative of making just k - 1 more offers, rather than k. That is, the seller could sweeten the current offer so as to advance the countdown by one period, inducing the buyer types who planned to accept next period to accept now instead, and so on. For k > 3 the condition needed to prevent the seller from switching to this alternative is Wk = [7 -- 6k- ']pk_ , + 8Wk_ , >_ [Y -- 5k- 2]pk_ , + Wk_ ,.
(10)
This is the basic optimality condition for the seller. It implies W k -- 1 ~-~ (~k -- 2 Pk - 1,
W k <-- 7Pk - 1"
(11)
Since the seller gains by shifting probability to earlier offers, the probability be reduced until condition (10) binds, implying that (11) holds as a pair
P k - 1 will
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of equalities, for k > 3. Thus 3k-2Wk-~-7Wk_l,
(12)
Wk--~7Pk_ 1"
For 2 < k < M - 1 these equations determine Wk and Pk recursively. In order to start the recursion, Pl must be determined: this is the smallest probability that will prevent the seller from pooling at the point where k = 2. The continuation value associated with pooling depends on the seller's assessment of the buyer's current type. Let (* be the screening threshold: this is defined as the solution of the equation (1 - 0 ( 7
- - 6) + 3 ( ( 7 o
-
1) =
0 + AU(~b( + 1 - tr) v.
(13)
This says that the expected value of pooling is the same as the value of making two offers, allowing for the fact that acceptance of the first offer reveals the high type, thereby increasing the seller's expected return from the next negotiation. The logic here is that 7,7o and the function U(o) are given, and these determine (*. Given the screening threshold, Pl is determined by the condition that the seller must be indifferent between pooling and screening when k = 2: ~* [pl + ~] = ~.
(14)
This gives W2 = [7 - 6]PI + 6[?o - 1]( = 609~, where the parameter co is defined by o~= [ ~ - I l Z *
+7o-1.
(15)
(16)
So
7k-2 Wk(?) -- 6 1/2~k- 3)k~0~.
(17)
Define the sequence nk as ~(7) = ~ -?T ~ -
1(?) -- ? k S - l/2k(~ + 3)
(18)
Then the optimal acceptance probabilities Pk are determined from the screening threshold (* and the variables (?, ?o) summarizing the continuation values as I4<. ?
p~(~,~) = ,+____!~; ~ _ ~ ( ~ ) ~ { .
(19)
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4.2. Determinin9 the number of offers For a given value of M, any reallocation of the probabilities Pi in favor of earlier offers increases the expected payoff. On the other hand, if too much weight is put on early offers then the screen will collapse as soon as these offers have been rejected. At each step in the recursion the proposed value ofpk is checked to make sure that it fits within the unit interval, and the recursion stops when this condition fails. Define k-1
/~k= 1-- ~ p~.
(20)
j=O
This is the largest probability that could be put on an initial offer which will be followed by k subsequent offers. The optimal number of offers must be such that this probability is insufficient to warrant extending the sequence from M to M + 1 offers. That is [7 - 6u]Pu + 6Wu < [Y - 6u - 1]Pu + Wra
(21)
d~ u < Wu.
(22)
SO
This, in combination with the reverse inequality warranting extension from M - 1 to M offers, implies W~
+Pl +P2 +"" +PM-2 < 1 - P o <---~ ( . / + Pl
d
M
M
k=3
k=3
+ P 2 + "'" +Pu-x,
Y YPl+ ~ Wk <--YEI--~]
(23)
which can be rearranged as " --~3 M y k - 3 1 1 - ~ ~ = 3 " k - 3Y (DOg (~l/2(k-2)(k-1)~ ( ( , ~ "~ (D(~R= t~l/E(k-~(k-1)"
(24)
Define 1
Z-~-
1,
1
Z*---~,
1.
(25)
Then M is determined by F l u - 3(7) <-
Z -Z* 09
< l-Iu- 2(Y),
(26)
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458
where M
n~(7) = E ~s.
(27)
s=O
That is, there is a system of weights ~k(~,6) that determines the number of offers and the probabilities assigned to each offer, for any initial assessment ( that the state is currently low. The acceptance probability for the initial offer is the value implied by continuation of the recursion (12), plus the remainder of the unit interval (this remainder being insufficient to support an additional offer). That is,
PM-1 =PM-1
= 1-~-pl-
~ Pk = 1-k= 1
--00~17M_40' ) -~
--'°Ez7"
(28)
The seller's optimal continuation value relative to the continuation value of the high-type buyer, is U
~nn= [7--d]PM-1 + 6W~-1 =Ey-d]¢~o[Z~ Z* = (o~[(y-d) Z-Z• co
l'Iu_ 4(y)] + 6y~u_ 4(yKo~
-
7I'Iu- 4(7) + dFIu- 3(')]-
(29)
This ratio can alternatively be written as VU. = ¢o~[ (7 _d)Z-Z*+d_f~_3(7) o,
]
= ( 7 - - d ) [ 1 - ~ . ] + ~eo[d-f~t- z(Y)],
(30)
wheref~(7) is defined as
fu(7) = 71"Iu- 1 -
6M+ 2(I-Iu -- 1)
(31)
with the convention that II_ 1 = 0 so thatfo = O.
4.3. The buyer's continuation values Consider the value of the buyer after M - 1 screening offers have been rejected, and the seller has therefore concluded that the buyer is currently in the low state.
J. Kennan/Japan and the World Economy 7 (1995) 447 472
459
The value of the buyer at this point does not depend on which type of screen has just ended. Define VL as the value of the low-type buyer that has just signed a contract, and define Vn as what the high type would be worth now, if it had rejected enough offers to fool the seller. The incentive compatibility constraint implies that this is also the value the high type would have had now, if it had accepted some previous offer and thus revealed that the current profit rate is high. Thus, for example, the value of the high type at the start of negotiations is ~ M - 1 VH' and this does depend on the type of screen, because the number of offers will typically be greater in a hard screen than in a soft screen; also the low type's value at the start is 6 M- ~ VL, which again depends on the type of screen. The low-type buyer gets nothing from the current contract, so the value of the buyer, as of the signing date, is VL = AL(K ) + AXd~[p(K)VL + {1 -- p(K)} VH],
(32)
where d a = 6 n ° - 1 is the discount factor from the beginning to the end of a soft screen, and AL(K ) is the expected present value of the sequence of pooling contracts: K-1
AL(K ) = ~ A*[1 - p(s)].
(33)
s=l
In Eq. (32), VL and Vn are functions of K (among other things). If the seller never screens, then the valuation of the low-type buyer reduces t o AL(OO).Note also that if the contract is of infinite length, VL is zero, as in the basic screening model. If the high type imitates the low type, the yield from the current contract is vn - vL = 1, so the continuation value of the high-type buyer at the signing date is Vn = AH(K ) + AKda [p(K){1 - pa(K)} VL + a(K)VH-],
(34)
where K-1
AH(K ) = 1 +
K-1
~ ASt~(s)= ~ AS[z*+Cbsz *] s=l
1 -- A r
s=O
1 -
fix
= z~ 1 -----A+ z~ - 1- - - - ~
(35)
Here fl = A~ is a discounted persistence measure that will prove important in determining the buyer's cost of revealing that profits are currently high. These equations can be used to determine VL and Vn, for any given value of K and M a. The most convenient way to do this is to first solve for the difference Vn - VL. The transition rule of the Markov chain yields p(s) + a(s) - 1 = tk~.
(36)
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Therefore
_ilk ~-da/3k[V._ VL], v . - VL--11_/~
(37)
which implies 1 -/3 K
VH- K = 1-1 - 13] 1-1 -/3Kda]"
(38)
Finally, by substituting for An(K) and Vn - VL in the equation for VHabove, and using the relation 1 - a(K) = (1 - 4/)z*, the continuation values for the buyer in each state may be written in terms of model's basic parameters as
v.
1 - AK
.
1 -/3x
Zn(l* - a)(1 - Ard.) ~-ZL(1 -- fl)(1 -- flrd.)
(39)
and 1 - AK
1 - fir
VL = z~ (1 -- A)(1 - AXd.) - z~ (1 - fl)(1 - flXda)"
(40)
These results cover various interesting special cases. For example, if the buyer is now in the low state, its value depends on the difference between A and/3: if the contract is very long, or if there is a high degree of persistence, this buyer is not worth much, as one would expect.
Solution method In equilibrium the continuation values must reproduce themselves. This gives a system of three equations in the variables (y, (*, 09). The "(* equation", defining the screening threshold, can be written as 0 U° 609(* = ~ a + A-~n"
(41)
The ";~ equation" is ~= l+0+Ad
VH
b~'n + AUb
(42)
VH
The "?o equation" can be written as c°-(~-1)
Z * = 0 ( 1 -AK)/(1V. -A)+ArU°vH"
(43,
In each of these equations, the U/V piece is continuous in the variables (y, (*. co), but VHjumps when K changes, and also when M o changes (unless K is infinite); in
J. Kennan/Japan and the World Economy 7 (1995) 447-472
461
addition, the 7 equation jumps when M b changes. These j u m p discontinuities are the main source of difficulty in attempting to solve the model. The three equations to be solved can be expanded as
0 + A(7 -
6e)~* = ~
1 +O+Ad 7=
a o) 1 - ~
l + A~°(oEd ° _ f o ( ? ) ] ,
P.+A(7_d b 1
VH
+ A K~ao)Ida __fa (7)],
(44)
where ~o = e ~ * + 1 - tr,
~'~ = p ( K ) ,
(45)
~b = 1 - a,
a n d f a = f M , - 3, etc. Aside from the discontinuities, all three equations are linear in o), and the second two are linear in ~*, or, equivalently, in Z*. So if the second two are used to solve for to and ~* in terms ofT, and these are substituted in the first equation, the result is a single equation in ~. This will be a polynomial with j u m p discontinuities. The problem with this approach is that the values of the integer variables (Ma, Mb, Ms, K ) are not known until the system is solved. Pooling
If the opportunity cost of a conflict is high enough, or if the probability of discovering the high type is low enough, the seller always makes a pooling offer. In this case even when the seller is most optimistic, with ( = 1 - a, the return from pooling now must exceed the return from screening, given unconditional pooling in the continuation game. That is, -
(1-a)(7-6)-Sa(Vo
1-fl
1 ) ~ < 0 1 _- A p- ,
(46)
where 7 and 70 are determined from Eq. (7), with K = 0o and U b = 0/(1 - A). Then 1 - Ap
P'H -
VH -- (1 -- A)(1 -- fl)'
a - fl
(47)
(1 -- A)(1 - fl)'
SO (1 - A ) ( 1 - fl) + 0 ( 1 - fl) + A d ~ ( a - fl)
7=
1 -- Ap
'
0 ( 1 - - fl) ~o = 1 + 1 --A-----~"
(48)
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If the seller will always pool in future then d b = 1, and the condition for optimal pooling now reduces to 1 -
Ap
1-fl
a
(49)
<~O.
1-a
If this holds with equality the seller is indifferent between pooling and screening when ( --- 1 - a. If the seller randomizes in this situation then 6 < d b < 1. This gives a region defined by (1
-
a A)l_a
1 -- Ap a - 1-fl l-a"
~< 0 ~< - -
(50)
Thus the equilibrium is completely determined whenever the first of these inequalities holds: the seller either screens randomly at (b or always pools. 5
The Coase property
The Coase property provides a useful reference point in interpreting the equilibrium, and it may also prove useful as a computational device. Let 6 approach 0 while A is held fixed, representing a situation where the seller's commitment power is negligible but the duration of each contract is fixed. In this case Eq. (8) implies that ui = Uo, for all i, so 7 = 70. Also z* A +
VH=I--
z*
1--fl
_
1 -
Ap
(l -- A)(1-- fl)
(51)
and 0 U= Ua=u o = 1-A
(52)
so that the seller's valuation is just the pooling value. Approximate equilibria
The Coase property can be used to find an approximate solution of the model. Suppose that the Coase property will apply in the next contract negotiation, but the seller has some commitment power in the current negotiation. In this case the screening threshold for the current contract is the value of ( that solves the equation (1 -
0(V*
-
6) + 6ff(7" -
1) = 7 " -
1,
(53)
5When K = oo and Mb< 3 the ratchet effectdoes not cause trouble: the ratchet condition (described below) holds for all ft.
J. Kennan/Japan and the World Economy 7 (1995) 447-472
463
which implies ~*
1
1 - Ap
7*
1-Ap+O(1-/~)'
7*(7* - 1)
to=
(54)
6
These parameter values can be used to generate an approximate equilibrium. Transient information When there is no persistence in the private information process, the model simplifies considerably, although the repeated contract structure still causes complications relative to the static case. The seller's continuation value does not depend on the outcome of the previous contract, so the right sides of the ( and 7o equations are identical. Subtracting one of these equations from the other gives a solution for to in terms of 7 and ~*; then subtracting the 7 equation from the equation and substituting for to yields 1
7(7-
~* = - , ~,
co = - c5
1)
(55)
The V equation can then be expressed as a monotonic function of ~, with jumps where M changes, so existence of a unique solution is guaranteed. This model is analyzed in detail in Kennan (1993). Numerical solutions The following ad hoc procedure was used to solve numerical examples. First, the problem can be reduced to two equations in two unknowns by using the 70 equation to solve for to as a function of(* and y. This equation is piecewise linear in to and since K is determined by (*, and U/VH is continuous in to, jumps discontinuities with respect to occur only where M, changes. Write the equation as
v.
+
-1
Z*+A~(~-d")
1-~
-R, to=0,
(56)
where R a ~ 1 - mK~a[da - f ~ ( 7 ) ] "
(57)
This is piecewise linear in to, and continuous except for the first term, which jumps when d a changes. The continuous part is decreasing in 09, since R a is positive. Also, VH is increasing in d a, so the first term jumps down when d" increases. As to increases the test ratio that determines M decreases, so when to passes a critical value M, decreases, da increases, and the first term jumps down. The upshot of all this is that the function is everywhere decreasing, with downward jumps where
J. Kennan/Japan and the Worm Economy 7 (1995) 447-472
464
~o crosses a threshold. If the function jumps across the axis then randomization over two adjacent values of M a is needed to get a solution. To compute the solution start by assuming Ma = 2, which means Z ° - Z* < o~. When the equation is evaluated at this threshold, the result is
O(1-Ar)/(1-A) v.
'Z,_Za
~~
[
(a]
+ AK~ 1 - ~
.
(58)
If this expression is positive, then the root must lie beyond this threshold, meaning M~ = 2. A negative value is possible because Z" exceeds Z*. In this case test the equation at the next threshold, which is Z ~ - Z * = e)II1(,)= ~° [ 1 + ~ 2 ]
(59)
meaning M. = 3, and continue until a positive value is found. This pins down M., and reduces the problem to solving a linear equation in o~. The solution is substituted in the ( and 7 equations, leaving two equations in the two unknowns
(~, (*). The next step is use the ( equation to solve for 7 as a function of (*. With (* given, each value of 7 generates values for all of the integer variables. Then the equation can be evaluated as a function ofT, and solved numerically. Finally, the result is substituted in the 7 equation, which is then solved numerically for (*. In order to handle the discontinuities, the numerical solutions are obtained using a homemade algorithm that starts with an interval where the equation changes sign, splits this interval in half at each iteration, and solves a linear approximation joining the end-points. Thus the length of the interval shrinks by at least a factor of 2" after n iterations. If the algorithm continues for a large number of iterations without finding a solution, the conclusion is that the equation jumps across the horizontal axis, so randomization is needed. An illustration of the computational difficulties is given in Fig. 2, which plots the (* and ~ equations as functions of Z* after substituting for ~o. The continuous parts of both equations are downward sloping (for the parameter values used in this example), and there is more action in the ~, equation, which jumps one way when Ma changes, and the other way when Mbchanges. Fig. 3 shows the final step of the solution process, in which the 7 equation is regarded as a function of Z*, after substituting a value of 7 that (numerically) solves the ~ equation at each value of Z*.
The ratcheteffect Screening is feasible in this model only if the ratchet effect is not too strong. The penultimate offer must be sufficiently attractive to induce the high-type buyer to forgo a sequence of K potentially lucrative pooling offers, submitting instead to
J. Kennan/Japan and the World Economy 7 (1995) 447-472
465
\
1.15-
\ 1.145gamma
1,14-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
zeta"
Fig. 2. y and (* equations. a hard screen in the next contract negotiation. But the amount needed to attract the high type might also be more than enough to attract the low type, in which case the screening equilibrium breaks down. For instance, Hart and Tirole (1988, Proposition 3) showed that in the fixed-valuation case (tk = 1) with single-period 1 (rental) contracts and ~ > ~, the union makes only pooling offers in equilibrium (unless the horizon is finite, and close at hand). More generally, the feasibility of screening depends on whether the discounted persistence parameter fl is sufficiently small.
466
J. Kennan/Japan and the World Economy 7 (1995) 447-472
J
0.001!
0.00'
L-
f 0.3
0.4
0.5
0.6
0.7
0.8
Z
-o.ooo~
Fig. 3. Reduced y equation as a function of Z.
Let V° denote the value of the low-type buyer accepting wl, given that the union incorrectly infers that the buyer is a high type. Then V ° = 0 - w 1 + Adb[(p Vt + {(1 - p}) Vn)].
(60)
Comparing this with Eq. (3) yields
~(EV. --
o ) = 1 "-F f ~ A d b E ( V H - VL] ). VL]
(61)
The low type's incentive compatibility condition can be stated as ~(VH - V°H)/> 6(VH - VL) and then, using Eq. (38), the feasibility condition for an
J. Kennan/Japan and the World Economy 7 (1995) 447-472
467
equilibrium with screening becomes
[6 - fldb] [1 -- fir] << [1 -- fl] [1 - daflx].
(62)
This can be written as 0 ~< hK(//) = 1 -- ~ --//(1 -- rib) +/~x(6 _ da) +//K+ l(d ~ _ rib).
(63)
This condition is difficult to check, since K, d, and db are not k n o w n in advance. A simple case arises when either there is no persistence across contracts or else the contract is infinitely long, so t h a t / / = 0: then the condition reduces to 6 _< 1, and the screening equilibrium is feasible. T h e fixed-valuation case is also transparent: h e r e / / = A, the union always m a k e s pooling offers after a screen reveals the low type, and always d e m a n d s v . after the high type is revealed, so K = oo, db = 0, and the condition reduces to ~ _ 1 - A, or 6 _< ½ for rental contracts (where T = 1 so ,~ = A). M o r e generally, the acceptable range f o r / / c a n be analyzed by first considering the case where K = 1, and then c o m p a r i n g hK(//) with hi(~~ ). F o r any K, hK(0 ) = 1 - 3 and h ~ ( 1 ) = 0 , while hi(~~) is a quadratic with roots at 1 and (1 - 5)/(do - db). Also,
hK + ~(//) -- h/c(//) = / ~ x ( / / _ 1)[6 - d,, + ~(a, - db)]
(64)
w i t h / / < 1 and 6 _> d, _> d b, so the function hK+ 1(//) lies below hK(//). The situation is illustrated in Fig. 4 for Ma = 2 and M b = 5. The admissible values o f / / a r e those below the smaller root of hK(//). If K is infinite, the ratchet condition is satisfied
The Ratchet Condition M..s-2, M_b-5
0.11114
\ 0.0~ 0.000
..: ? ~ . •,~... ~
.-.____
" . "...., -0.0~
"4
". . . .
-0.004 0.0 0.1 0.2 0.8 0.4 0.5 0.6 0.7 0.8 0.9 1.0 8 --K-1
"Km2--K-5
F i g . 4.
J. Kennan/Japan and the World Economy 7 (1995) 447-472
468
when fl is below (1 - 6)/(1 - db). Reducing K allows fl to increase slightly, but the upper bound of(1 - 6 ) / ( d a - d b ) means that fl must be small unless strikes are very short. Thus there may be a tradeoff between allowing the model to explain long strikes, implying virtually no persistence across contracts, or allowing it explain linkage across contracts, with strikes that are too short to match the data.
5. Examples Some examples of the above results are shown in Table 2. The examples show that equilibria of the model can be found that match the intuitive description given at the outset. The seller goes through regular cycles of optimism and pessimism, with unsuccessful screens being followed by a depressed pooling phase, ending with a very short screen, while successful screens generate optimism about the gains from screening in the next negotiation. The length of the pooling phase, and the maximal duration when the union screens, respond to changes in the transition probabilities, and to changes in contract length. The examples suggest that if there is substantial persistence, the contract length must be very long in order to avoid the ratchet effect. The discount factor 6 is Table 2 Numerical examples
vL vH wo 6 p a T d~ z* A 7 ~* ~o w1 K VL
Vn Ma Ua Mb Ub
h(fl)
A
B
C
D
E
F
1/11 12/11 0 0.9975 0.98 0.9 1200 0.88 0.833 0.0496 1.0906 0.8844 0.10356 0.091 10 0,0011 1.0467 2 0.0957 23 0.1304 0.0002
1/I 1 12/11 0 0.9975 0.95 0.9 1200 0.85 0.667 0.0496 1.0906 0.8829 0.1038 0.091 3 0.0027 1.0467 2 0.0957 23 0.1303 0.0002
1/11 12/11 0 0.9975 0.95 0.6 1200 0.55 0.889 0.0496 1.0925 0.9047 0.1028 0.0928 4 0.0027 1.0307 2 0.0957 10 0.1035 0.002
1/11 12/I 1 0 0.9975 0.95 0.6 900 0.55 0.889 0.0496 1.0946 0.8841 0.1068 0.0927 ~ 0.0062 1.0676 1 0.0952 10 0.1021 0.0012
1/11 12/11 0 0.9975 0.1 0.9 150 0 0.1 0.687 1.1537 0.8667 0.1778 0.0978 1 1.76 2.76 16 0.4583 16 0.4583 0.0025
0.0934 1.0934 0 0.998 0.1 0.9 150 0 0.1 0.687 1.156 0.865 0.18 0.1
1 1.767 2.767 15.45 0.491 15.45 0.491 0.0025
469
J. Kennan/Japanand the World Economy 7 (1995) 447-472
intended to refer to offers spaced one week apart, as in Kennan and Wilson (1991), so a 3-year contract corresponds roughly to T = 150. Thus the contract length in column E is realistic, but this example assumes no persistence, so that the equilibrium involves screening in each negotiation. Examples A - C show that sequences of pooling offers are made when the degree of persistence is relatively high, but the contracts in these examples last about 24 years.
5.1. Randomization by the uninformed seller
Randomization by the informed buyer is a familiar feature of static screening models in which the distribution of buyer types is discrete. A novelty of the repeated screening model is that randomization by the uninformed seller may also be required in order to reach an equilibrium. During the countdown, the
1,I~5
1.1~
1.1~5
Gamma 1.12
1,1565
Fig. 5. Randomization.
1,157
1.15~
1.1~
470
J. Kennan/Japan and the World Economy 7 (1995) 447-472
seller is indifferent at each stage between making k or k - 1 more offers and it is important that this tie is broken in favor ofk. But if the parameter values are just right, the seller is indifferent at the outset between M and M + 1 offers, and in this case any random mixture of these choices will do equally well, from the seller's point of view. Equilibrium requires that the seller chooses the right mixture in this situation. An example of this is in column F of Table 2. Fig. 5 shows the ? equation for this example, as described in Section 4, with the (unique) solution lying on the vertical section of the curve.
5.2. Experienced bargainers do not fight It is generally found that conflicts in labor negotiations are most severe when the union is negotiating its first contract. Column D in Table 2 contains an example in which there is no private information in the long run, because the probability of the high state, according to the stationary distribution of the Markov chain, is too low, relative to the cost of conflict, to warrant screening. Nevertheless, if the union is initially optimistic for some reason there will be a screen in the first contract, and if the high state is discovered there will again be a screen in the next contract, and so on until the low state is discovered after which the union makes pooling offers in all subsequent contracts.
Acknowledgments This paper was presented at the Ninth Annual Japan-US Technical Symposium, New York University, March 1994. The paper grew out of a joint research project with Robert Wilson, to whom the author is grateful for many enlightening discussions. The National Science Foundation provided the research support.
References Blume, A., 1990, Bargaining with randomly changing valuations, Working Paper 90-22, University of Iowa. Borland, J. and J. Tracy, 1988, The determinants of expected contract duration and early renegotiation: An empirical analysis, Yale University, unpublished. Card, D., 1987, A longitudinal analysis of strikes and wages [-draft of Card (1990b)], Princeton University, April. Card, D., 1988, Longitudinal analysis of strike activity, Journal of Labor Economics 6, 147-176. Card, D., 1990a, Strikes and bargaining: A survey of the recent empirical literature, American Economic Review 80(2), 410-415. Card, D., 1990b, Strikes and wages: A test of an asymmetric information model, Quarterly Journal of Economics 105, 625-659. Cramton, P.C., 1984, Bargaining with incomplete information: An infinite horizon model with two-sided uncertainty, Review of Economic Studies LI(4), (167), 579-594.
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Cramton, P. C., 1992, Strategic delay in bargaining with two-sided uncertainty, Review of Economic Studies 59, 205-225. Cramton, P. C. and J. S. Tracy, 1992, Strikes and holdouts in wage bargaining: Theory and data, American Economic Review 100-121. Cramton, P. C. and J. S. Tracy, 1994a, The determinants of U.S. labor disputes, Journal of Labor Economics 12(2), 180-209. Cramton, P. C. and J. S. Tracy, 1994b, Wage bargaining with time-varying threats, Journal of Labor Economics 12, forthcoming. Fischer, S., 1977, Long-term contracts, rational expectations and the optimal money supply rule, Journal of Political Economy 85, 191- 205. Fudenberg, D., D. Levine and P. Ruud, 1985, Strike activity and wage settlements~ UCLA Working Paper 249, revised September 1985. Fudenberg, D. and J. Tirole, 1983, Sequential bargaining with incomplete information about preferences, Review of Economic Studies 50, 221 247. Gray, JoAnna, 1978, On indexation and contract length, Journal of Political Economy 86, 1 18. Harris, M. and B. Holmstrom, 1987, On the duration of agreements, International Economic Review 28, 389-406. Harrison, A., 1993, Strike and contract data for postwar canada, McMaster University, (unpublished draft.) Harrison, A. and M. Stewart, 1989, Cyclical variation in strike-settlement probabilities, American Economic Review 79, 827-841. Hart, O., 1989, Bargaining and strikes, Quarterly Journal of Economics 104, 25-44. Hart, O. and J. Tirole, 1988, Contract renegotiation and coasian dynamics, Review of Economic Studies 55, 509-540. Hayes, B., 1984, Unions and strikes with asymmetric information, Journal of Labor Economics 2, 57-83. Hicks, J. R., 1932, The theory of wages (Macmillan, New York). Ingram, P., D. Metcalf and J. Wadsworth, 1991, Strike incidence and duration in british manufacturing industry in the 1980s, Working Paper No. 88, Centre for Economic Performance, LSE, April 1991. Joskow, P. L., 1987, Contract duration and relationship-specific investments: Empirical evidence from coal markets, American Economic Review 77, 168-185. Kennan, J., 1986, The economics of strikes, in: O. Ashenfelter and R. Layard, eds., Handbook of Labor Economics, Vol. II (Elsevier, Amsterdam). Kennan, J., 1993a, Repeated contract negotiations with transient private information, presented at the Society for Economic Dynamics and Control, Annual Meetings, Nafplio, June 1993. Kennan, J., 1993b, Information cycles in repeated contracts, presented at the NBER Summer Institute, July 1993. Kennan, J. and R. Wilson, 1989, Strategic bargaining models and interpretation of strike data, Journal of Applied Econometrics 4, $87-S 130. Kennan, J. and R. Wilson, 1990a, Can strategic bargaining models explain collective bargaining data? American Economic Association Papers and Proceedings, 80(2), 405-409. Kennan, J. and R. Wilson, 1990b, Theories of bargaining delays, Science 249, 1124-1128. Kennan, J. and R. Wilson, 1993, Bargaining with private information, Journal of Economic Literature 45-104. Kennan, J. and R. Wilson, 1991, Screening models of bargaining with private information: An empirical application, unpublished draft. McConnell, S., 1989, Strikes, wages, and private information, American Economic Review 79, 801-815. Riddell, W. C., t979, The empirical foundations of the phillips curve: Evidence from Canadian wage contract data, Econometrica 47, 1-24.
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Riddell, W. C., 1980, The effects of strikes and strike length on negotiated wage settlements, Relations Industrielles 35(1), 115-120. Sobei, J., 1989, Durable goods monopoly with entry of new consumers, UC San Diego, (unpublished.) Sobel, J. and I. Takahashi, 1983, A multi-stage model of bargaining, Review of Economic Studies 50, 411-26. Tracy, J. S., 1986, An investigation into the determinants of U.S. strike activity, American Economic Review 76, 423-436. Vincent, D., 1990, Bilateral monopoly, non-durable goods and dynamic trading relationships, Northwestern University, CMEMS #832, revised January 1990. Vroman, S., 1989, A longitudinal analysis of strike activity in U.S. manufacturing: 1957-1984, American Economic Review 79, 816-826.