E U R O P E A N U N IV E R S I T Y IN S T IT U T E , F L O R E N C E
DEPARTMENT O F ECONOM ICSE U I W O R K I N G P A P
A STRATEGIC MODEL O]
WITH INCOMPLETE
by V , Douglas G * University o f Pittsburgh and University o f PennsylvaniaThis is the revised version o f a paper presented at the Workshop on Mathematical Economics organized by the European University Institute in San Miniato, 8-19 September 1986. Financial support from the Scientific Affairs Division o f NATO and the hospitality o f the Cassa di Risparmio di San Miniato are gratefully acknowledged.
BADIA FIESOLANA, SAN DOM ENICO ( F I )
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No part of this paper may be reproduced in any form without
permission of the author.
(C) Douglas Gale
Printed in Italy in November 1987 European University Institute
Badia Fiesolana - 50016 San Domenico (Fi) -
Italy
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I . INTRODUCTION
An unresolved problem in macroeconomic theory is how to reconcile excess supply with competition in the labor market. One way to resolve this longstanding problem is to assume that changes in the wage alter the quality of the workforce. For example, a link between price and quality might be explained by an efficiency wage hypothesis. Another explanation might be that the wage is a screening device. The second approach has been followed in an important paper by A. Weiss [7]. The argument is simple. Workers are assumed to be heterogeneous. They differ both in their productivities and in their outside options. (Outside options can be opportunities to work in other industries or simply profitable uses of leisure time). A worker will accept a job only if it is at least as attractive as his outside option. Workers are assumed to know their own productivity and outside options but firms are not. The crucial assumption is that productivities and outside options are positively correlated. High productivity workers tend to have better outside options. This is what allows the wage to serve as a screening device. A reduction in the wage reduces the average quality of the workforce by encouraging the most productive workers to quit. This reduction in productivity costs the firm more in revenue than it gains from the reduction in the wage bill. So in this case a wage cut actually makes the firm worse off. Weiss uses this idea to explain why firms do not reduce wages even when there are job queues. It can also explain why firms simultaneously hire and fire workers.
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These results are suggestive of a theory of unemployment. What remains to be done is to incorporate them in a theory of labor market equilibrium. Weiss' arguments all relate to the behavior of a single firm facing given distributions of workers. They do not encompass the determination of equilibrium prices and quantities in a multi-firm market. This is unfortunate since the essential problem is to account for unemployment as an equilibrium phenomenon. In this paper I describe a model of labor market equilibrium incorporating the central features of Weiss' paper. The objective is to define the market allocation process as an extensive form game and to characterize the equilibria. In subsequent work these results will be applied to the analysis of unemployment in the short and in the long run.
There is no uniquely correct way to model competition in a market characterized by incomplete information. The presence of incomplete information introduces a genuine strategic element into the interaction of agents. The outcome of this interaction may well depend on the detailed modeling of the allocation process. The model developed in this paper borrows ideas from the literature on search and bargaining [1,2,5]. The model belongs to a class of strategic models of exchange that share three central features. First, all exchange occurs between pairs of agents. Second, agents search randomly for trading partners. Third, the terms of trade are determined by the agents themselves. When a pair of agents meets, one of them will make an offer and the other agent then has the opportunity to accept or reject the offer. This seems a very
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natural way to model an allocation process in labor markets; but it is not the only way. To define the allocation process we have to specify the agents who make up the market, the matching process that brings them together and the rules governing the interaction of matched pairs of agents. We begin with a description of the agents.
Agents are divided into two classes called workers and firms. There is a continuum of workers and a continuum of firms. Each firm wishes to hire at most one worker and each worker seeks at most one job. All firms are assumed to be identical but workers are not. Workers are distinguished by their productivity and the value of their outside options. Each worker is characterized by his type 0. Without loss of generality we can take the set of types to be the unit interval and the distribution of types to be uniform on the interval. The productivity of a 0-type is denoted by A (0) and the value of his outside option by B(0). Both A and B are assumed to be strictly increasing in 0, that is, more productive workers have better outside options. Note that a worker's productivity is assumed to be independent of the firm that hires him. Let L > 0 denote the measure of workers in the labor market and N > 0 the measure of firms. The labor market is completely characterized by the numbers L and N and the functions A and B.
The next step is to define the matching process. Matching takes place at an infinite sequence of dates indexed by t = 0,1,2,... All the agents are present in the market at date 0
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and they remain until they have traded or exercised their outside options. At that point they are said to leave the market. An agent who has left the market cannot re-enter the market and does not participate in any way. At each date t, the agents who remain in the market are randomly matched. Each worker has a probability 0 < < 1 of being matched with a firm. Each firm has a probability 0 < < 1 of being matched with a worker. The matching process is assumed to satisfy three properties. First, the process is symmetric in the sense that each worker (resp. firm) has the same probability of meeting every firm (resp. worker). Second, it is serially independent in the sense that the matching probability of any agent is independent of what has happened to him at previous dates. Third, there is no aggregate uncertainty. The cross-sectional distribution of matches is the same as the probability distribution. In particular, the proportion of workers (resp. firms) matched at date t would be a^ (resp. /S^) . The matching probabilities are endogenous to the extent that they depend on the numbers of workers and firms and the distribution of workers' types at each date. But since there is a large number of agents each individual agent regards them as parameters.
Finally, we have to define the rules governing price formation. All offers are assumed to be made by firms. When a firm and worker are matched, the firm will announce a wage offer u>. Then the worker has three choices. He can accept the offer, he can reject the offer or he can take his outside option. If the worker either accepts the offer or takes his outside option the game is
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over for him. If he rejects the offer he continues to search, that is, he waits until the next date when he will be matched again. For the firm, the game ends only if his offer is accepted. Otherwise he continues to search, that is, waits until the next date when he will be matched again. All agents are assumed to maximize their expected utility. A worker's utility is either the wage he receives or the value of his outside option if he takes it. A firm's utility is the difference between the productivity of the worker he hires and the wage he pays. Agents who remain in the market forever get zero utility.
The equilibrium concept used to analyze this game is essentially the subgame perfect equilibrium. An agent's strategy is a sequence of functions {f } that map his history h^_ into a feasible action a^ - f^h^) at each stage t of the game where he has to move. An equilibrium strategy must be a best response to the strategies of other agents at each stage t for every possible history h^. But in applying this criterion, we take as given the matching probabilities a^ and and the distribution of workers' types at each date. What is important for the results is that workers respond optimally to every possible offer and not just to the offers made in equilibrium.
The extensive form game sketched above is one way to implement an allocation in the labor market defined by {L,N,A,B}. The purpose of analyzing the extensive form game is to suggest a concept of market equilibrium, that is an allocation mechanism that depends only on the parameters of the market {L,N,A,B} and not on
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the details of the extensive form. The main result of the paper is an equivalence theorem relating the equilibria of the extensive form game and the equilibria, suitably defined, of the labor market. To define the market equilibria some more notation is needed. For any wage to let II(to) denote the expected profit from offering to to a randomly chosen worker. II(to) is defined by
n(w) - |
[
a(
0) - w]d«.
The worker is assumed to accept the offer if and only if /S(0) < to. For any to let ft(to) denote the expected profit from hiring a randomly chosen worker at to. That is, ft(to) is the expected profit from repeated offers of to made to independently chosen workers until one accepts. Then
ft(u) - II(w)/Pr(B(0) < u ) if Pr(B(0) < u ) > 0 = 0 otherwise.
A
Suppose that II has a unique maximum at to^, that II has a unique
A A
maximum at to., and that II and II are both positive on the interval M
( b t o ) and negative on (to , b. ) , where b. = B(0) for 9 - 0,1.
0 max ° max 1 6
Suppose further that 0 < to., < to., < to < B(l). (Later,
rr M M max
assumptions will be made on the primitive data of the market to ensure that these conditions are satisfied). Figure 1 illustrates the assumed properties of the profit functions.
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7
The wage levels u>„ and w would be chosen by monopolists who
M M
could make single and repeated offers, respectively. These offers might also be made by non-atomic firms since, as is well known, the monopoly price may be attained in a competitive market when only one side of the market can make offers. However, we can also define a market-clearing wage in the usual Walrasian sense. If the auctioneer calls out a>, firms know it will only be accepted by 6-types for whom B(0) < to. The expected profit conditional on w
A
and the market-clearing assumption is II(w). The Walrasian demand function is denoted by D(w) where
D(w) A - o if n(w) < o A £ [0,n] if n(w) - o A - n if n(o>) > o.
The Walrasian supply is
S(u>) - L B ^ [max {min {b., w) , b^}]
for any w. There is a unique, market-clearing wage at which D(u>) — S(u>) .
With these definitions we can state the main result of the paper. [The reader is warned that this is a rather loose statement that omits some assumptions and qualifications introduced in Section II]. Let {co^} be the sequence of wage offers made in an equilibrium of the extensive form game. Then co^ = u>* for all dates t and
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(a) i f w». > ox. > t o „ t h e n t o * —Aox. — M M C ---A * M (b) i f “ M > o>_ > ax, t h e n to ■ 0 M. A * ■ "c (c) i f U . > ox. > t o . , t h e n to e[ax,, a>_ 1 .
— C M M --- 1 M' C J
T h e m o d e l w i t h a c o n t i n u u m o f t y p e s y i e l d s a v e r y n e a t statement of the theorem; unfortunately it is also hard to analyze. For this reason I study a discretized version of the model sketched above, one that has only a finite number of types. The discretized model is an approximation to the continuous model in an obvious sense and yields the desired result for the continuous model in the limit as the number of types becomes very large. There is one drawback to the discretized model, however, and this explains why I want to view it only as an approximation to the continuous model. The problem is that, generically, the case (b) equilibria mentioned above do not exist. Consider the situation depicted in Figure 2 and suppose that ax, > a>_ > ax,. Then the theorem claims that
M O M
* *
o> - a) so all 9 - types with 9 < 9 will prefer to to their outside options and all 9 types will be indifferent. The equilibrium allocation requires all 9 - types with 9 < 9^ to accept jobs while some types accept jobs and some take their outside options. Such an allocation is acceptable to workers since the 9^-types are indifferent. But firms are not. Since 9^-types are the most productive among those actually hired, any firm would be willing to offer o> + c for some c > 0 small enough to ensure that all 9
-c c
*
types accept. Thus to = to^, is not an equilibrium and in fact no equilibrium exists under the maintained assumptions.
This problem does not arise if the measure of agents with
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types 6 < is exactly equal to N. We can always approximate the continuous model (in which the existence problem does not occur) using a discrete model with this property. But since the property is not generic in discrete models, the theory developed in this way is best thought of as applying to continuous models.
Another caveat concerns the class of equilibria that is examined. Only symmetric equilibria in which agents choose pure, history independent strategies are considered. I do not consider these restrictions as very onerous and I conjecture that the same results could be proved without them, though it might not be easy. More restrictive assumptions are the following. I assume that firms make serious offers, that is, if it is optimal for the firm to make an offer that would be acceptable to some type of worker, it does so. Second, I assume that if it is optimal for a worker to accept a job and he strictly prefers the job to his outside option, then he accepts the job. And conversely when the roles of jobs and options are reversed. These last assumptions apply to cases where agents are indifferent between taking a particular course of action today and doing the same thing at some future date. It considerably simplifies the proof to assume that they do not "hang around" gratuitously in such circumstances. Whether these restrictions rule out interesting equilibria is not known.
It must be emphasized, of course, that none of these restrictions requires agents to behave in suboptimal ways. These are restrictions on equilibria not on equilibrium strategies.
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Every agent's behavior is optimal within the class of all feasible strategies and not simply those satisfying these restrictions.
10
What the main result shows is that Weiss' conjecture is consistent with equilibrium in a fully specified strategic model. Case (a) is precisely the case in which firms do not choose to reduce wages even though they could do so and still hire a worker with certainty. In this sense we have provided firmer foundations for the view that markets may not clear when the wage acts as a sorting device.
It may seem that equilibrium analysis is superfluous since it confirms the conclusions that Weiss based on analysis of a single firm's behavior. In particular the detailed extensive form game may seem unnecessary. I think this conclusion would be wrong for several reasons. In the first place, as the characterization theorem shows, other cases can arise. (The reader should note that the values of ww and depend only on the functions A and B. On
M M
the other hand, can vary between B(0) and the lesser of to.. and
C Max
B(l) as the ratio of L and N changes, holding A and B constant. Thus all of the cases in the theorem arise in a given market for different values of L/N.) Without a model of equilibrium one cannot say which particular case will arise or what factors determine the regime that obtains in equilibrium.
So a model of equilibrium seems to be essential. The reason why we need a "dynamic" model, i.e., one in which bargains are struck over "time", is to allow for the possibility that information revealed at one stage may be used at another. As
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Oliver Hart [4] has pointed out in reference to the paper of Gul and Sonnenschein [3], most models of signalling are not robust if recontracting is allowed. A static model without recontracting may seriously misrepresent the working of a market with incomplete information. In Weiss' model the fact that a worker leaves his job signals that he is of higher quality. Why doesn't the firm exploit this fact, first reducing the wage and then, if the worker attempts to leave, making a higher offer as he goes out the door? Of course this would not be an equilibrium if workers anticipated the higher offer with certainty. But a mixed strategy could discriminate among workers. Something similar could, in principle, happen in our model. Suppose the pattern of wage offers were like the one in Figure 3. A worker with a high option like 0 will leave immediately. If the first wage offer is serious (i.e., if any worker will accept it) the worker with a low option like 6^ must accept it. However, there may be workers with intermediate options like 0^ who are willing to wait for an offer like <x>^. The difference is that if 6^ does not get the high offer he has his outside option B(0j) to fall back on; whereas 0^ would have to make do with the lower u> if he missed the chance of w.. Now if
<x> i
separation occurs, so that the population in period 2 contains a higher proportion of workers like 0^ than the population in period
1, it may be optimal for firms to offer a higher wage in period 2
than in period 1, thus justifying the workers' beliefs. The really hard part of the proof is showing that this kind of separation cannot occur.
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Of course, we have only shown that separation does not occur in this model and under this set of assumptions. The value of this kind of exercise is not to claim that this is "reality" but simply to clarify what sort of assumptions are necessary to get a particular kind of result. The recent paper by Rubinstein and Wolinsky [6], for example, shows how important assumptions about information sets and the matching process may be in guaranteeing a determinate equilibrium.
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L3
II. THE DISCRETIZED MODEL
In this section I describe in detail the discretized version of the model sketched in Section I. There is assumed to be a finite number (n + 1), say, of types. The types are indexed by 9 = 0,h,2h,...,l where h ^ - n. The initial distribution of types is assumed to be uniform so Pq(0) = (n + 1) ^ for any 9, where Pq(0) denotes the proportion of 9 - types in the population of workers at date 0. As before L and N are the measures of workers and firms, respectively, in the market at date 0 and it is assumed that L > N > 0 . A( 9 ) and B(0) represent the productivities and outside options, respectively, of 9 - types for any 9 . 11(a)) and ft(o)) are defined as in Section I. The following assumptions are maintained throughout.
(A. 1) A(0) > B(0) > 0 and A( 9 ) and B(0) are strictly increasing in 9 .
(A.2) a)w = arg max 11(a)), a\. = arg max ft (a)) and ox, < o\ _.
M M M M
For some types and #M , B(0M ) - B(#M ) - wM and < 0^.
(A.3) (a) II(B(»>) <(>) n(B(« + h)) as B <(>) 6 ^, (b) ft(B(»)) <(>) ft(B(« + h)) as S <(>)
Finally, the discrete structure is chosen so that at some price a) = B(0^) we get exact market-clearing:
(A.4) For some type 9^ and to = B(5^), ft(o)) > 0 and L V * c > ' N
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14
This assumption implies the existence of an interval of market clearing prices Q - {B(0 ) < to < B(ff + h) } . The first two assumptions require no comment. (A.3) says the functions II and ft are strictly quasi-concave on the set {B (0), B(h).... B(1)}. It is needed for the usual reasons. (A.4) is needed to ensure the existence of equilibrium in the extensive form game. (This issue is discussed in detail in Section IV). Note that (A.4) is violated generically in the discrete case. This is one reason why I want to think of the discrete case as only an approximation to the continuous case, where the existence problem does not arise.
Matching
The following notation is used to describe the matching process.
L - measure of workers active in the market at date t; = measure of firms active in the market at date t;
pt (0) = proportion of active workers who are of type 9 at date t;
- probability of a worker being matched at date t; = probability of a firm being matched at date t.
It is assumed of course that Lq = L > 0 and = N > 0. The matching probabilities are defined by
(1) o t - A min {Lt ,Nt )/Lt (t = 0,1,2,...) and (2) - A min (Lt ,Nt )/Nt (t - 0,1,2,...).
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All workers have the same probability of being matched, independently of their histories. Similarly all firms have the same probability of being matched, independently of their histories. The matching process is assumed to be symmetric in the sense that each worker or firm has, respectively, the same probability of being matched with someone from a particular group of firms or workers. Finally, it is assumed that there is no aggregate uncertainty. In particular, the proportion of workers who are matched at date t is and the proportion of firms who are matched at date t is If a worker and firm are matched at date t the probability that the worker is a 0-type is denoted by Pt (0).
A worker or a firm who is unmatched at date t must remain in the market until date t + 1. It is assumed that not all agents are matched at any date t. That is,
(3) 0 < A < 1.
Consequently, at any date there is always a positive measure of firms and workers in the market:
(4) Lt > 0, N > 0 (t - 0,1,2, . . .).
The probability of a worker meeting a firm or a firm meeting a worker is therefore also well-defined and positive:
(5) 0 < <*t < 1, 0 < 0 < 1 (t = 0,1,2,...).
Since workers and firms can exit from the market but cannot enter L > L . and N > N .
t t+1 t t+1 for all t = 0,1,2,... This implies that
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the measures of active workers and firms converge to and respectively. The measure of workers of type 0 at date t can be denoted by Lt (0) where of course 1,^(0) * LtPt(0). By the same reasoning Lt(0) > Lt+^(0) for every 0-type and t - 0,1,2,... and Lt (0) - 1^ (0) say.
Workers
The wage offered at date t is denoted by and the sequence of wage offers by u = {Wq,w- ,...}. An offer is made to a worker if and only if he is matched. Define a random variable m^ for every date t by putting
(6 ) mt
-{
10 with probabilitywith probability 1 - a^.The interpretation obviously is that m^ - 1 if a worker is matched and m^ = 0 if he is unmatched at date t. The highest payoff the worker can obtain at date t is max{u>t ,B(0)Jm^. If he is unmatched he gets nothing; if he is matched he can get the wage offered or his outside option, whichever is greater. The worker's problem is
to choose an optimal stopping time, r, to solve the problem
(7) max E [max{w ,B (0)}m ]
T T T
For any date t and any type 0 define
(8) [max{u>^ , B (0)}m^ ] .
V t+1(0) is the payoff a worker of type 0 will get in equilibrium if he has not already accepted a wage offer or taken his outside
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option by date t. If a worker is unmatched he must decide whether to accept the firm's offer, take his outside option or reject the offer and remain in the market. Clearly he chooses the maximum of B(0) and V t+^(0). His behavior is described by a decision rule dt(u>) that gives this decision at date t as a function of the wage offer. Formally, a decision rule is a function d^: !R+ -*■ {accept, reject, option} and it is optimal if
(9) dt («)
"accept" if u) > Vt+^(0), B(0); "reject" if V ^(0) - B(0); "option" if B(0) > V t+^(0), w
-Firms
At each date t a matched firm chooses a wage offer For any type 9 the probability that 9-types will accept is denoted by P t (w t >0)* The probability p^_ is defined by
(10) p t (u>,0)
1 if w > max lv t+1(0). B(0)} 0 if u < max (Vt+1(0), B(0)>.
[We appear to assume that in cases where w - B(0) > V t+^(0) the worker always accepts. But this must be true in a perfect equilibrium. Since the type $ with B($) - w is always the most profitable of those workers who might accept, it would be optimal for the firm to offer + c if $ refused for some sufficiently small c > 0.] Then define the profit functions
(11) n t (w) - X d P t (w,0)[A(0) - u>]
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18
and
(12) - n t M / x 0 pt(w,5)
for any co such that 2 pt (co,0) > 0. Put ^ ( w ) = ftt (w) = ^ otherwise. Let p t(u>) ■ 2 p t(co,0) denote the probability that an offer of co will be accepted at t. The sequence of wage offers made by a firm is denoted by u ■= {co^}. Given a sequence of offers co the probability that a firm remains in the market at date t is denoted by 7t (w ) and defined by
t-1
(13) y ( » ) - n (1-p (to ) 0 )
t - k _ o K k K
The firm's problem is to choose a sequence to of offers to solve the optimization problem:
oo
(14) W = Max 2 y M 0 n (co )
U k-0
For any date t let denote the maximized expected utility of a firm that remains in the market after date t. Then
co
Wt+1 - Max S V - N W V / V S ) k-t+1
for every date t. An optimal sequence co is one that solves (14). From the principle of optimality, co is an optimal sequence if and only if, for every date t,
(15) wt e arg max (ftt (wt)pt(wt) + Wfc+1 (1 - P t (wt)))
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The representation of the firm's behavior in (15) is extremely useful in what follows. Some facts that are immediately apparent are noted below.
LEMMA 1: Let to be an optimal sequence of offers and suppose that P t(a>t.) > 0 for every date t. Then
(a) ftt(wt) ^ wt+1 and w t = wt+1 if and only if * W t +1 ’ (b) W t > w t+-^ implies > W f o r every s < t;
(c) =» w t+| implies = ^s+i f°r everY s > t;
(d) ^ t(w ) - if w ^ and ftt (<*>) - f°r ©very u>
if - wt+1.
Proof: (a) If the first inequality were false the firm would do better, according to (15), to choose w * 0 and pt (o>t) = 0. The second fact follows because W^_ is a proper convex combination of ft (to ) and W . .
t v t' t+1
(b) Suppose to the contrary that = ... * W^_ > w t+-^ • Such a case must exist if (b) were false. Then 6^(0?^) = ft^(w^) = ... = ft^w^) * W^_. Then with some positive probability the firm has an offer accepted before date t+1 and gets W^_; with some positive probability he does not have an offer accepted before t+1
and then he gets . Since is the expected value of these two cases, W Q < W t , a contradiction.
(c) This is an immediate corollary of (b) .
(d) Suppose, contrary to the first claim, that to > to^ and ftt (w) > ftt (u>t) . Then pt (u>) ^ P t (w t) and ftt (w t;) - imPly that
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(16) â t(u>)pt(w) + W tl(l - P t(u>)) > ftt(ut)pt(«t) + w t+1<1- - Pt (“ t)) contradicting (15). To prove the second claim, note that
> fit - W t+1 ^mP^^-es t^iat P t(u)) > 0 so that again we have (16) and a contradiction. ||
LEMMA 2: Let w be an optimal sequence of offers. Assume that at some date t > w t+l anc* ^'tyPes acceP t cot at t (resp.
at t+1) if and only if B(0) < (resp. B(0) < c*>t+^) . Assume also that - ^t+1 ' ^hen ^t+1 = W t+2 *
Proof : If $ = max{0|B(0) < w t+-^) then optimality requires and so ^ t+i(^) = Vt+2^^* Consequently any 0 < $ will accept an offer of + t at date t for any £ > 0. That is,
p t(“ t+i + £) - W W and
(17) i i - nt(»t+1 + «) - n t+1(«t+1).
Similarly, since ^ V t+2 ’ any ^ _t:yPe b^at accepts at t will
accept u + £ at t + 1. Then p t (a>t) = Pt+l^w t + anc*
(18) lim n t+1(a>t + e) - 1 1 ^ ) .
£“*0
Now (15) implies that for any £ > 0,
(19) nt(wt)
+(l
- Pt(^t))wt+1 *nt(wt+i
+o
+(l
- Pt(«t+1 +0)W
t+i and <2 0 > n t+ l (“ t+ l> + (1 - V “W > W t + 2 * n t (“ t + l + O + (1 - P t(»t + 0 ) Wt+2 '©
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Taking limits in (19) and (20) , substituting (17) and (18) and rearranging gives
- pt+i<“ t+i» wt+2 - W - W W - (P t (“ t> ■ p t+ i (“ t+i))W It is clearly not optimal to choose to^ > unless p t (a>t) >pt+ (wt+ ) so W « > W . a s claimed. II
t+2 t+1 11
As an immediate corollary we see that under the conditions of Lemma 2, W . - W « implies to < to , .
t+1 t+2 v t t+1
Equilibrium
An agent's strategy is a sequence of functions or decision rules that tell him what action to take at each stage of the game where he has to move, as a function of his complete history up to that stage. For example, a firm's strategy tells him what wage to offer if he is matched as date t and in principle it may depend on the previous offers he has made, whether he was matched or unmatched in particular periods, the responses of the workers he met and so forth. In fact, conditioning on past history can never increase an agent's utility in this game. This is because an agent's history provides no information about the agents he will meet in the future. In any case, I am going to assume that agents do not condition on their past history. This does not appear to be essential for the results but it considerably simplifies the notation. In addition, I am going to assume that agents of the same type choose the same strategy. That is, there will be one strategy for all firms and for each type of worker. Finally, all
t+1'
agents are assumed to use pure strategies. In other words we are
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looking for a symmetric equilibrium in pure, history independent strategies. An equilibrium in this sense is described by a sequence of numbers >N t > ^ tIo and ^y strategies {d^C* ;0 ) t-Q and {w } * The array {Lt , N t , Pt , d^ ( •; $) ,u>^} is an equilibrium if and only if
(a) (w^) is optimal for a firm given {L^.N^.P^.d^O ;0O);
(b) (dt (* ;^))j is optimal for a 0-type at date T given {L^.N^, and any offer to at T;
(c) {L ,N ,P } are consistent with the matching process, the strategies { to } and { ( •;<?)} and the initial conditions Lq = L and NQ - N.
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III. CHARACTERIZATION OF EQUILIBRIUM
In this section I take a fixed equilibrium, as described in the preceding section, and analyze its properties. Recall that and are strictly positive for every t and -♦ > 0 and N -*■ N > 0 . The first proposition shows that at least one side of
t CO
the market clears.
PROPOSITION 1: min {L^.N^} - 0.
Proof: Suppose, contrary to what is to be proved that > 0 and N > 0 . Then a -* a > 0 and there exists at least one type $ such
CO t ® y
that L (£) -> L ^ d ) > 0. Since strategies are symmetric (all ft- types use the same decision rule) at any date t the measure of $- types leaving the market is either or 0. Now for some small £ > 0 and all sufficiently large t, L ($) - L < £ < . This implies that for all sufficiently large t, = Lt+^($) . Then V t+^($) = 0 < B($) for all t sufficiently large since all types in the market at t remain there forever. However, a necessary condition for this to be true is V t+1(0) > B(#) since a positive fraction of types are matched at each date. This contradiction establishes the desired result. || Proposition 1 tells us that we need only consider the following three, mutually exclusive cases:
(A) > 0 - N^; (B) - 0 - N^; (C) - 0 < N^.
These cases may be thought of as representing situations of excess
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supply, market-clearing and excess demand respectively. In the remainder of this section each case will be examined in turn.
Case A .
In this case it is obvious that a -* 0 and 3 -* 3 - A > 0.
t t ®
While /3^ A ensures that each firm will be matched infinitely often with probability one, it is not so clear that the probability of a worker being matched at least once converges to zero as t approaches infinity. The next proposition establishes this fact.
PROPOSITION 2: Lim_ 8 (1 - a ) = 1.
T^° t-T C
Proof: As we have seen, for some $, L ^ ^ ) > 0. For an agent of type $ there are three possible outcomes. Either (a) he accepts a wage offer or (b) he exercises his outside option or (c) he remains in the market forever. Let denote the probability of a $-type remaining in the market forever given that he is in the market at date t. Let denote the probability of a ^-type accepting a wage offer at some date given that he is in the market at date t. Now
At < N t/Lt (ft) - 0 and 1 - r < [L (#) - Lj h ] / b < h - 0 t t CO t Then v t + 1 ( £ ) < At+1sup{us |s > t+1) + ( 1 - r t ) B ( S ) -> 0 < B($)
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as t -* « since {u> } is bounded. Since V . ($) < B($) for all t
s t+1
sufficiently large, a $-type will exit whenever matched for all sufficiently large t. But then Lt ($) -► > ®.||
As a simple corollary we see that for some T, for all t > T and for all types 9
B(0) > V t+1(#).
This means in particular that all 6-types will accept any offer co > B(0) for t sufficiently large.
PROPOSITION 3: For some finite value of t, w = u> = inf{co }.
t s
Proof: The proof is by contradiction. If the proposition were false there would exist an infinite number of values of co^ and, in particular, for some t > T it is true that * B( 9 ) for any 9. For example, B(0) < < B(0 + h) . But V t+^(0) < B(0) for t > T and so a firm could reduce without changing the probability of acceptance (strictly speaking, without changing the set of types that accept . This implies that was not optimal for the firm, a contradiction.!
PROPOSITION 4: If = w and $ = max{0|B(0) < then Pt (0) = P^C^ for all 9 < § .
Proof: For any 9 < and all t it is optimal for 9 - types never to exercise their outside options since B(0) < . Then for any date t and type 9 < ft, v t+1(0) = V t+^(£). In particular, for s < t if w is a serious offer then V .( 9 ) < a> (otherwise no one would
s s+1 s
accept the offer). In equilibrium any 9 - type who is matched at
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date s will accept the offer of to . For either B(0) < to or 9 = 6 and B($) “ w s • In the latter case $ must accept; otherwise it would be optimal for the firm to offer u>s + £ for some £ > 0
sufficiently small, contradicting the definition of equilibrium. But if, at each s < t, every matched 9 < ft accepts, then the only 9 - types with 9 < ft remaining in the market at t are those who have yet to be matched. The distribution of types is therefore uniform on the set {9 < ft) as claimed. ||
Propositions 3 and 4 are essential to the central result in the characterization of Case A. This result is proved next.
PROPOSITION 5: For any date t, u > wu . t M
Proof: According to Proposition 3, to^ - to for some finite t. Consider two cases: (a) t > T and (b) t < T.
Case (a) . § accepts to^ if and only if 9 does for all 9 < ft. (cf. the argument in Proposition 4). Since t > T we must have » - B ( h . (cf. the argument in Proposition 3) . For any value of 9 we have
V t+1(<? + h) " V t+1(<0 + B(* + h) ' B(*}
implying + h) < B(0 + h ) . If a firm were to offer + £ at date t, for any £ > 0 it would be accepted by all 9 < $ + h. Suppose, contrary to what we want to prove, that to = B($) < v Then for c > 0 sufficiently small we have ft(B(^)) < ft(B($ + h) + £) and so (21) = ft(wt) - ft(B(#)) < fl(B($ + h) + £) < ftt(B(ft + h) + £).
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The last inequality follows from the fact that + h) > P^_($) * Pt (0) for all 9 < But (21) contradicts the optimality of the firm's offer since it could increase fl^ by raising contrary to Lemma 1. So in this case w > ww as claimed.
Case (b) . We can ignore the case where = B($) since this is essentially the same as Case (a) . Then B($) < < B(ft + h). Choose t to be the largest such value. I claim that s > t implies w > B(ft + h ) . If not, then for some s > t, B($) < w < w <
s t s
B($ + h) . We have u> > w since u> - w and by the choice of t,
s t t —
o>s * a>t . But this cannot be optimal since ^ s+^(^) —
so u>s can be reduced without affecting the decisions of any types of workers. This proves the claim and it follows that it is optimal for all types 6 < $ + h never to exercise their outside options after date t. It is immediate that “ V t + 1 ^ + **) for all 9 < $ + h. Since v t+^(^) - wt an offer of B($ + h) + z must be accepted by all 9 < $ + h for any c > 0. Suppose again, contrary to what is to be proved, that w < Then for sufficiently small c,
ft (w ) - ft(w ) < ft(B(ft)) < ft(B(ft + h) + e) < ft (B(ft + h) + e) ,
contradicting the optimality of w . ||
PROPOSITION 6: W > ft(» ) for any date t.
PROOF: It is immediate from -*■ X and -*■ 0 as t -*■ « that a firm can attain a payoff ft(w) for any w < w. It is simply a matter of offering u> at each date when matched. The offer will be
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accepted by any 6 such that B(0) < u and with probability one such a match eventually occurs. Then W is at least as great as the maximum of f!(w) for u < w, that is . ||
PROPOSITION 7: For any date t, - u^.
Proof: Suppose that > w - Since Pq is uniform,
* V v * w i * w t+i a *<“«>•
But wA > ww implies fl(w.) < , a contradiction. Thus u>A = u>x.
U M U M U M
and all matched agents at date 0 will leave the market. Then P. = P_ and by induction it can be shown that w = £>.. for all t.||
1 0 t M
It is clear that in order to have L > 0 - N , the premises of
co co
Case A, it must be true that u>^ > supft^. Case B .
This case shares some of the properties of Case A but the fact that V t+^(0) does not converge to zero requires a somewhat different strategy.
PROPOSITION 8: For all t sufficiently large and some B($) < < B($ + h ) .
Proof: The proof is by contradiction. Suppose to the contrary that, for some $ and arbitrarily large values of t, to^ < B($) and
> B($). Now $ cannot be certain of getting > B($) ; otherwise no one would accept < B(0). On the other hand he is sure of getting at least B(^) because market-clearing requires that at any date there is probability one that he will be matched with a firm at some future date. The positive probability of getting > B($)
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then ensures that V > B(0) for all t. So $ never exercises his outside option. But since he cannot be sure of getting
> B($) this means that, with positive probability, he remains in the market forever, a contradiction of market-clearing. ||
Using the same sort of argument we have the following corollary. Corollary. Either c*>t * B($) for all t sufficiently large or
> B($) for all t sufficiently large.
PROPOSITION 9: The sequence is eventually constant.
Proof : Suppose that {to^} is not eventually constant. By the corollary to Proposition 8, B(^) < w < B($ + h) for arbitrarily large values of t. Since > B($) for all sufficiently large t an argument used earlier in Proposition 4 shows that
for all 9 < 'è and t sufficiently large. Then for sufficiently large t, < B($ + h) implies that and all 9 ■< § accept the offer if matched. Now consider two cases.
(a) Suppose that < B($ + h) for all t sufficiently large. Then
“ t - ‘ “ t+i- But if "t > “t+i then v t+i (^ < a contradiction, so - wt+l ^or C su^^ic ^-ent^-y large.
(b) Now suppose that u>t = B($ + h) for arbitrarily large values of t. Then for some (large) t, and = B($ + h) . Then Lemma 2 implies that w t+-^ “ W t+2 anc* S° ’ Lemma 1, {W^} is eventually constant and ft^co^) = w t+| for all sufficiently large t. But if m , « < B($ + h) and ft (c*> ) — ft. (w.) then
S t s s t t
cû = So far large t, {a>t } is constant whenever < B($ + h Now we have a contradiction, because = constant when
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< B($ + h) and - B($ + h) infinitely often implies v t+i(2 ) > V ||
From Proposition 9 we have to^_ -* Since every worker will be matched infinitely often with probability one if he remains in the market we have V -(0) > w for all 0 and t. Then to. > to for
t+1 oo t ®
all t, that is, - u for all t sufficiently large. Then using the arguments of Propositions 5 and 6 we have and W > n(a> ) and then, as in Proposition 7, one can show that
t ®
to > tox, for all t. Then it is clear that L = 0 = N requires
t ao M oo co
the set of market-clearing prices. Hence, we have proved the following result.
PROPOSITION 10: For any t, w - to
t co Ü
Case C .
The premise that - 0 < N^ implies that -* A > 0 as t-+®. A worker who remains in the market will be matched infinitely often with probability one.
PROPOSITION 11: to -* to as t ■» « and u> > to for any t.
t ® t ®
Proof: I claim that for any 6 and any t, V t+^(0) — li® sup to t-KO
For any e > 0 , to^ + c > lim sup infinitely often. Since -* A > 0 a worker who accepts only offers to > lim sup - c will succeed in getting such an offer with probability one. Since e is arbitrary, V ^(0) — lim SUP <*>t - It Is immediate that u>t -*■ and to > to for all t . II t « 11
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PROPOSITION 12: For all t sufficiently large, to^_
-Proof: Suppose the claim were untrue. Since to -* to and u> > to
--- r r t oo t «
for all t there exist infinitely many values of t such that > to^ for all s > t. For any such value of t and some 0, = B(0). If not, let $ denote the largest value of 0 accepting and we have c*>t > max{B(^), V t+^($)}, contradicting the optimality of to^. Since there is only a finite number of values of B(0) there cannot be an infinite number of values of t such that to > to for all s > t. So
t s (w^) is eventually constant after all. ||
PROPOSITION 13: For any t, w - w .
Proof: > 0 since it is always possible to make a profitable trade. But this fact together with > 0 = L implies that lim x (1 - a ) = 1. The probability of remaining unmatched
T t=T
forever after T converges to 1 as T -* *>. Hence W t \ = 0. From Lemma 1 we have that for every t. Let t be the largest date such that to > to , = to . Then a t t 0 accepts to if and only
t t+1 ® t
if B(0) < . (The usual argument applies in the case where
B(0) - u t ) . Then Pt - P ^ . and
-II t+^(^t+l) ' Lemma 2, a contradiction.!
PROPOSITION 12: < w < inftwcfh,}.
Proof: The second inequality is straightforward given = 0 < N^. To see the first suppose to the contrary that Then for all t, II ( u j =
n(co
) < II(co..) = IL(a\_). Then for t sufficientlvt t t M t M large
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(22) n(«t) + (i - Pt (^))wt+1 < n t(uM ) + (i - Pt ( V )Wt+i
since \ 0, where $ is the largest value of 6 accepting u>t . (22) contradicts the assumed optimality of and proves that ||
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References
[1] K. Binmore and M. Herrero, "Matching and Bargaining in Dynamic Markets, Parts I and II" London School of Economics (1983 mimeo.
[2] D. Gale, "Limit Theorems for Markets with Sequential Bargaining", Journal of Economic Theory (forthcoming).
[3] F. Gul and H. Sonnenschein, "Uncertainty Does Not Cause Delay". Princeton University (1985) mimeo.
[4] 0. Hart, "Bargaining and Strikes", M.I.T. (1986) mimeo.
[5] A. Rubinstein and A. Wolinsky, "Equilibrium in a Market with Sequential Bargaining" Econometrica 53 (1985) 1133-1150.
[6] A. Rubinstein and A. Wolinsky, "Decentralized Trading, Strategic Behavior and the Walrasian Outcome" University of Pennsylvania (1986) mimeo.
[7] A. Weiss, "Job Queues and Layoffs in Labor Markets with Flexible Wages" Journal of Political Economy 88 (1980) 526-538.
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B _1(Nd/L) FIGURE 2
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to> Bf V B ( « j ) b
(«l )
(t-1) (t=2) FIGURE 3©
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WORKING PAPERS ECONOMICS DEPARTMENT
85/155: François DUCHENE Beyond the First C.A.P.
85/156: Domenico Mario NUTI Political and Economic Fluctuations in the Socialist System
85/157: Christophe DEISSENBERG On the Determination of Macroeconomic Policies with Robust Outcome
85/161: Domenico Mario NUTI A Critique of Orwell’s Oligarchic Collectivism as an Economic System 85/162: Will BARTLETT Optimal Employment and Investment Policies in Self-Financed Producer Cooperatives
85/169: Jean JASKOLD GABSZEWICZ Paolo GARELLA
Asymmetric International Trade
85/170: Jean JASKOLD GABSZEWICZ Paolo GARELLA
Subjective Price Search and Price Competition
85/173: Bere RUSTEM
Kumaraswamy VELUPILLAI
On Rationalizing Expectations
85/178: Dwight M. JAFFEE Term Structure Intermediation by Depository Institutions
85/179: Gerd WEINRICH Price and Wage Dynamics in a Simple Macroeconomic Model with Stochastic Rationing
85/180: Domenico Mario NUTI Economic Planning in Market Economies: Scope, Instruments, Institutions 85/181: Will BARTLETT Enterprise Investment and Public
Consumption in a Self-Managed Economy 85/186: Will BARTLETT
Gerd WEINRICH
Instability and Indexation in a Labour- Managed Economy - A General Equilibrium Quantity Rationing Approach
85/187: Jesper JESPERSEN Some Reflexions on the Longer Term Con sequences of a Mounting Public Debt 85/188: Jean JASKOLD GABSZEWICZ
Paolo GARELLA
Scattered Sellers and Ill-Informed Buyers A Model of Price Dispersion
85/194: Domenico Mario NUTI The Share Economy: Plausibility and Viability of Weitzman's Model 85/195: Pierre DEHEZ
Jean-Paul FITOUSSI
Wage Indexation and Macroeconomic Fluctuations
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85/196: Werner HILDENBRAND A Problem in Demand Aggregation: Per Capita Demand as a Function of Per Capita Expenditure
85/198: Will BARTLETT Milica UVALIC
Bibliography on Labour-Managed Firms and Employee Participation
85/200: Domenico Mario NUTI Hidden and Repressed Inflation in Soviet- Type Economies: Definitions, Measurements and Stabilisation
85/201: Ernesto SCREPANTI A Model of the Political-Economic Cycle in Centrally Planned Economies
86/206: Volker DEVILLE Bibliography on The European Monetary System and the European Currency Unit. 86/212: Emil CLAASSEN
Melvyn KRAUSS
Budget Deficits and the Exchange Rate
86/214: Alberto CHILOSI The Right to Employment Principle and Self-Managed Market Socialism: A Historical Account and an Analytical Appraisal of some Old Ideas
86/218: Emil CLAASSEN The Optimum Monetary Constitution: Monetary Integration and Monetary Stability
86/222: Edmund S. PHELPS Economic Equilibrium and Other Economic Concepts: A "New Palgrave" Quartet 86/223: Giuliano FERRARI BRAVO Economic Diplomacy. The Keynes-Cuno
Affair
86/224: Jean-Michel GRANDMONT Stabilizing Competitive Business Cycles 86/225: Donald A.R. GEORGE Wage-earners' Investment Funds: theory,
simulation and policy
86/227: Domenico Mario NUTI Michal Kalecki's Contributions to the Theory and Practice of Socialist Planning 86/228: Domenico Mario NUTI Codetermination, Profit-Sharing and Full
Employment
86/229: Marcello DE CECCO Currency, Coinage and the Gold Standard 86/230: Rosemarie FEITHEN Determinants of Labour Migration in an
Enlarged European Community 86/232: Saul ESTRIN
Derek C. JONES
Are There Life Cycles in Labor-Managed Firms? Evidence for France
