Wage bargaining with incomplete information in an unionized Cournot oligopoly

European Journal of POLITICAL ELSEVIER

European Journal of Political Economy Vol. 13 (1997) 353-374

ECONOMY

Wage bargaining with incomplete information in an unionized Cournot oligopoly Vincent J. Vannetelbosch

*

CORE and IRES, University of Louvain, 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium Received 1 March 1995; revised 1 November 1996; accepted 1 December 1996

Abstract We develop a model of wage determination with private information in an unionized oligopolistic industry and we use it to compare the outcome of collective bargaining under two different bargaining structures - one in which a single wage is bargained at the industry-level and one in which the wages are bargained at the firm-level. The bargaining process is described by Rubinstein's alternating-offer bargaining model with two-sided incomplete information about the bargainers' discount rates. Iterative elimination of strictly conditionally dominated strategies for games with incomplete information leads to bounds on the payoffs which may arise in the game and to bounds on the wage agreements that may be made. Moreover, the perfect Bayesian equilibria payoffs and wages conform to these bounds. The main results are: firm-level wage outcome is not necessarily lower than industry-level wage outcome; potential inefficiency (hence, possible strike activity) is larger when bargaining takes place at the industry-level. JEL classification." C78; J50 Keywords: Wage-levels; Bargaining; Dominance; Incomplete information; Strikes

I. Introduction The Belgian labour market has b e e n characterized b y a shift in the level of wage negotiations. Since the late seventies and b e g i n n i n g eighties (see Table 1),

* Tel.: + 32-10-474321; fax: + 32-10-474301; e-mail: [email protected]. 0176-2680/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PHS0176-2680(97)00009-8

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Table 1 Collective agreements and strikes Year

Sectorial agreements

Firm-level agreements

Number of strikes

1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987

392 453 637 556 738 500 401 518 503 400 402 243 186 277 220 346 192 512

38 69 54 50 66 51 44 46 120 225 341 480 624 1862 1076 3253 1436 1989

84 92 95 79 124 98 98 99 47 50 30 48 53 21 15 9 5 8

negotiations have become more and more decentralized (firm-level negotiation). Previously, wage negotiations were mainly conducted at the industry-level. This shift coincides with a substantial decrease in the number of strikes having wages for motive. Thereby it may be interesting to study the relationship between the level of negotiation (or bargaining structure), the wage outcome and the strike activity, i.e. to understand how institutional features such as the bargaining structure affects the outcome of the negotiations. In this paper, we will provide some insight into the nature of collective bargaining and the impact of different bargaining structures in unionized oligopolistic industries. We assume the following game structure: first unions and firms bargain over the wage level according to institutional features (centralized or decentralized bargaining), and then firms compete (Cournot competition) by choosing their level of output (hence, their level of employment). This game structure captures the fact that wages are often fixed long-term contracts while firms' output decisions are variable over a shorter period of time. We adopt Rubinstein (1982) alternating-offer bargaining model with two-sided incomplete information about the bargainers' discount rates for describing the wage bargaining process. Indeed, bargaining may be interpreted as a process of exchange of offers and counteroffers necessitated by opposite preferences and by initial differences in information known to the negotiators separately. The assumption of two-sided incomplete information, along with the assumption that the firms and the unions are rational, does not rule out the possibility of strikes (delay may be required to convey private information) in equilibrium.

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The paper applies Watson (1994) results on Rubinstein's alternating-offer bargaining game with two-sided incomplete information to a model of wage negotiation in an unionized oligopolistic industry. Iterative elimination of strictly conditionally dominated strategies (IECDS) for games with incomplete information leads to bounds on the payoffs which may arise in the game and to bounds on the wage agreements that may be made. Moreover, the perfect Bayesian equilibria (PBE) payoffs and wages conform to these bounds. Recent work on the relationship between the bargaining structure, the wages and the strike activity includes Davidson (1988), Horn and Wolinsky (1988), Cheung and Davidson (1991). Cheung and Davidson (1991) have used a noncooperative bargaining model with one-sided incomplete information and a finite horizon to examine the relationship between the bargaining structure and the strike activity. They have focused on the rate at which information is transmitted during the bargaining process, a factor that could explain why strikes occur at equilibrium. Their investigation on how various levels of information transmission affect strike incidence and duration leads to the following conclusion: when a single bargaining agent represents the interest of workers at more that one firm, incentives are created that lead to a greater expected level of strike activity. In their analysis, Cheung and Davidson (1991) have assumed that the firms produce in separate markets in order to highlight the role of information transmission. But, when firms produce in related product markets, wage settlements create spillover effects (by altering the firms' relative competitive positions in the product market) that have implications for the outcome of negotiations. Wage spillover effects have been studied by Davidson (1988), Horn and Wolinsky (1988), but in a complete information framework so that strikes cannot occur in equilibrium l. In our paper, we go beyond the analysis offered in Davidson (1988), Horn and Wolinsky (1988), Cheung and Davidson (1991), by generalizing their results to an N-firm, N-union, infinite horizon, two-sided incomplete information bargaining setting. This setting enables us to investigate for different bargaining structures how private information as well as spillover effects across payoff functions created by contract settlements affect the wages, the level of employment and the strike activity.

1 Davidson (1988), Horn and Wolinsky (1988) have considered a duopolistic industry in which two different bargaining structures are mainly studied: one in which the workers of each firm are represented by separate and independent unions and one in which one union represents all the workers in the industry. Two market cases have been examined: either the firms produce substitutes (Davidson, 1988; Horn and Wolinsky, 1988) or they produce complements (Horn and Wolinsky, 1988). The main results are the following ones. When the final products are substitutes, then bargaining at the industry level leads to higher wages, lower profits for one main reason. If a firm agrees to pay a higher wage, its competitive position in the output market will be weakened and its competitors will respond by increasing output and employment. This externality across firms is internalized when an industrywide union forms. Therefore, the industrywide union holds out for a higher wage. When the products are complements, these relations are reversed.

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The main results of the paper follow. First, the decentralized wage outcome is not necessarily lower than the centralized wage outcome. This result is due to the assumption of two-sided incomplete information. Indeed, when the game becomes one of complete information then the centralized wage outcome is always higher. Secondly, potential inefficiency (delayed agreement, i.e. strike activity) is larger when bargaining takes place at the industry-level. The intuition behind our results comparing centralized and decentralized wage bargaining has to do with the wage spillover effects. These spillover effects are internalized partially by collective negotiation at the industry-level, which raises the potential payoffs but, in expanding the payoff set, also increases the scope for delay and inefficiency. There is a bound on delay in equilibrium which means, whatever the bargaining system, that an agreement is reached in finite time. Finally, only potential inefficiency at the firm-level depends on the number of firms in the industry and vanishes as the number of firms becomes large. The paper is organized as follows. In Section 2, the model is presented. The Coumot game in the oligopolistic market is solved assuming that the wages have already been determined. Section 3 describes the wage bargaining game (i.e. the bargaining process) and solves this game for the centralized bargaining system (i.e. wage negotiation at the industry-level). Section 4 is devoted to the wage bargaining game for the decentralized bargaining system (i.e. wage negotiation at the firm-level) and analyses the relationship between the bargaining structure and the wage outcome. Section 5 analyses the relationship between the bargaining structure and the strike activity. Section 6 concludes.

2. The model Consider a market for a homogeneous commodity produced by N > 2 identical profit-maximizing firms, denoted n = 1. . . . . N. Let q, denotes the quantities of the commodity produced by firm n. Let P ( Q ) = a - bQ be the market-clearing price when aggregate quantity on the market is Q = Y'.~=~q,. More precisely, P(Q) = a - bQ for Q < a / b , and P(Q) = 0 for Q > a / b , with a, b > 0. Assume that the finn is producing under constant retums to scale with labour as the sole input, i.e. q, = I n, where 1, is labour input. The total cost to finn n of producing quantity q, is q,,w,. The general price level is normalized to unity so that w~ is the real wage in firm n. Associated with each firm there is a continuum of identical workers who supply each one unit of labour with no disutility. We define ~n as the expected real income of a worker who loses his job in the firm n. Assume that ~, = ~ for all n = 1. . . . . N; where ~ may be interpreted as the unemployment benefit. Firm n's workers have identical preferences which are represented by the following von Neumann-Morgenstern ( v N - M ) utility function:

u,(w,,~,c.)=c~v(wn)+(l-c,)v(~

),

n=l .....

N

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357

where v(.) is a strictly increasing and concave function. The variable c n is the probability of being employed in the firm to which the worker is initially allocated. In each firm the workforce is represented in the wage bargaining process by a utilitarian union. We assume that workers are risk-neutral and we normalize to unity the continuum of workers who supply labour to each firm. Hence, union n's utility can be written as:

Un(Wn, W, ln) = I n w n q - ( 1 - l n ) w ,

/7=1 .....

(1)

g

Interactions between the product market and the bargaining level are analysed according to the following game structure. In stage one, wages are bargained at the firm-level (decentralization) or at the industry-level (centralization)2. In stage two, Cournot competition occurs: firms simultaneously choose their quantifies to produce, which determines their levels of employment, the industry output and the market cleating price. The model is solved backwards. In the last stage of the game, the wage levels have already been determined (through collective bargaining) and the N firms compete by choosing simultaneously their outputs (and hence, employment) to maximize profits with price adjusting to clear the market. A strategy of firm n is a quantity choice qn E ~ + . R+N is the set of strategy profiles. Let ~.U = (Trn)n ~ N be a list of all firms' profit functions 7rn:R+N ---) • which depend on the strategy profile q ~ R+N. The profit received by firm n for each strategy profile q ~ R+N is given by:

7rn(q)=qn(a-wn-bQ),

n=l

.....

N

We denote q-n (qm)m~ N\{n} as the strategies chosen by the other firms. This Cournot game 3 of complete information is denoted by C(N, ~N+, ,n.lV). To solve the Cournot game we use the Nash equilibrium concept. The reaction curve of firm n, n = 1. . . . . N, is given by: =

a-Wn_--bQ_ n

f~(Q-n) =

l

2b

if

Q_n<

1 ~(a

wn) -

1

if

Q_n>~(a-Wn)

N where Q _ , = (Em= l q m ) -- qn" Let qff denote firm n's monopoly output, aM_ n =-~N M M * ( m= 1qm ) -- qm" We denote by qn firm n's Nash equilibrium output of the game

2 Note that we define centralization (decentralization) as the inter-union and inter-employercooperation (noncooperation)in wage setting. 3 It is common knowledge that all wage contracts have already been settled and are known by all firms when firms compete on the product market.

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C(N, R~, 7rN). The strategy profile q* ~ ~+N is a Nash equilibrium if, for each firm n, q~* solves:

max 7rn(qn, q*n) = O
max [ q n ( a - w n - b Q * n - b q ~ ) ] ,

n=l ..... N

O
(3) where Q*_,, = ~-~m ~ N\{n}q;" Assuming an interior solution to the Coumot competition game, the unique Nash equilibrium of this stage game yields: Q * ( w l . . . . . Wu) q ; ( w , . . . . . WN)=

Na-W b(N+ 1)' a-Nw,

+ W_ n

b ( U + 1)

(4) ,

n = 1. . . . .

U

(5)

where W ~U,= IW" and W_, = W - w,. The Nash equilibrium output of a firm (and hence, equilibrium level of employment) is decreasing with its own wage and the number of firms in the industry, while it is increasing with other firms' wages and total industry demand. But a notion like Nash equilibrium implicitly assumes a substantial amount of coordination of players' behaviour. "... in some situations the Nash concept seems too demanding. Thus, it is interesting to know what predictions one can make without assuming that a Nash equilibrium will occur" (Fudenberg and Tirole, 1991, p. 45). Therefore, it is also interesting to solve the Cournot game using a weaker notion of equilibrium. That is, for characterizing the behaviour of the firms, we use the notion of iterative elimination of strictly dominated strategies (IEDS). IEDS assumes less exogenous coordination and derives predictions using only the assumption that the rules of the game (i.e. the strategy spaces and the payoffs) and the rationality of the players are common knowledge. In Appendix A we give the normal-form definition of strict dominance and we derive the following result. Except for the duopoly case ( N = 2), the Cournot game C(N, N, 7r N) is not solvable by IEDS. In what follows, to overcome the problem of an undetermined outcome with IEDS for the Cournot game C ( N > 2, ~u+, 7r N), we assume some coordination of firms' behaviour such that the firms produce the unique Nash equilibrium output of the game. Note that, as the product market becomes large, the Nash equilibrium of the game C(N, R N, ~. N) approaches the perfectly competitive equilibrium. By a large market we mean that, for a given and fixed demand, the number of firms becomes large. But if the number of firms is large, then IEDS does not approach the perfectly competitive equilibrium (all quantity choices between zero and the monopoly output survive IEDS). In the first stage of the game, firms and unions negotiate the wage level foreseeing perfectly the effect of wages on firms' decisions concerning employ=

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ment. To provide some insight on the impact of different bargaining structures in oligopolistic industries, we consider two bargaining structures: centralized and decentralized wage settlement.

3. Centralized wage bargaining At the industry-level, workers are represented by a single unions representative, which we call the Union. The Union's objective function is to maximize the sum of local unions' utilities. The Union negotiates the industry wage level with the employers representative, which we call the Firm. The Firm's objective function is to maximize the sum of individual firms' profits. These bargainers correctly anticipate the effect of wages on subsequent Cournot competition game. When the negotiation is centralized, a uniform wage is set by industrial associations for all firms.

3.1. The bargaining problem There are two bargainers - - also called players - - (the Union and the Firm) who must agree on a wage w from the set X. X is the set of feasible agreements: X = {w ~ E l0 < w < a}. The players either reach an agreement in the set X, or fail to reach agreement, in which case the disagreement event E occurs. The two bargainers have well defined preferences over X U {E}. The v N - M utility of a firm n for an agreement w is its profit

7r(w, l* ( w ) ) = l* ( w ) [ a - w - bNl* ( w ) ] while that of an union n is the total amount of money received by its members

u(w, ff~, 1" ( w ) ) = l* ( w ) w , + ((1 - l* ( w ) ) ~ If the two parties (the Union and the Firm) fail to agree, then a firm n obtains a profit of zero and an union n receives ~, so that the Firm's and the Union's disagreement points are, respectively, zero and N-ft. The utility function of a local union is unique only up to a positive affine transformation. For the sake of presentation, we rewrite local union's utility function:

u(w, ~, l* ( w ) ) = l* ( w ) [ w - - ~ ] such that the Union's disagreement point shifts from N-~ to zero. Therefore, the v N - M utility of the local union n for the agreement w is a--w

.(w,

b ( N + 1)

(w -

(6)

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360

II Firm's utility

(0,0)

U Unioffs utility Fig. 1. The bargaining set.

while that of a local firm is

a-w]

2

W e assume that there is free disposal, so that the set o f possible utility pairs U n i o n - F i r m that can result f r o m a g r e e m e n t is

Y-

O, N b ( N + l ) ( W - ~

) , O,

b

N+I]

]1 [ w ~ X

This bargaining set Y is depicted in Fig. 1. This is a c o m p a c t c o n v e x set, which contains the d i s a g r e e m e n t point d = (0, 0) in its interior 4. Thus (Y, d ) is a bargaining problem.

3.2. The wage bargaining outcome 3.2.1. How do the players reach an agreement? The negotiation is m o d e l l e d as a n o n c o o p e r a t i v e bargaining game. The bargaining process is described by Rubinstein (1982) alternating-offer bargaining model. The bargaining procedure is as follows. The bargainers can take actions only at periods in the infinite set { 1, 2 . . . . }. These bargainers m a k e alternatively w a g e

4 Let ~ = ½(a + ~); ~, is the wage which maximizes the Union's utility and should be chosen by the Union if the Union had the power to settle the wage unilateraly. From Eqs. (6) and (7), it is immediate that the frontier of the feasible utility profiles is a parabola and thereby concave; for all w ~ [0, ~] we have that au/aw > O, for all w ~(ff~, a] we have that au/aw < 0, while for all w ~ X we have that azr/aw < O. But once we assume free disposal of the resources, we obtain the bargaining set depicted in Fig. 1, where on its boundary we have that au/aqr < O.

V.J. Vannetelbosch / European Journal of Political Economy 13 (1997) 353-374

361

offers, with the Union making offers in odd-numbered periods and the Firm making offers in even-numbered periods. The bargaining game starts in period 1 with the Union proposing an agreement (an element of X). At period 2, the Finn either accepts the offer or proposes a counteroffer. The game ends when one of the bargainers accepts the opponent's previous offer. The Union is assumed to strike in every period until an agreement is reached. Both players are assumed to be impatient. The Union and the Firm have, respectively, a discount factor of ~u and ~f, where ~i ~ (0, 1) for i = u, f. We assume that all unions have the same discount factor 8u and all firms have also the same discount factor ~f. This assumption is supported by the homogeneity of the players ( N identical firms and N identical unions). Let H i be the set of histories after which player i has the move, for i = u, f. Let H = H u U He. Let h = O be the history at the start of play. A pure strategy 5 of player i is some function si:H i -o A which maps each possible history after which player i has the move into an action. In each period, the action space for the player who moves is A = X U {accept}, except in the first period, when the Union cannot accept a previous offer from the Firm. Let S i be the set of strategies for player i, i = u, f. Let S = Su × Sf. Each strategy profile s ~ S induces an outcome which specifies agreement at some date or irrevocable disagreement. Payoffs in the wage bargaining are given as functions of the players' strategy profile according to the v N - M utility functions U:S -~ R (for the Union) and I I : S --* (for the Firm). For any strategy profile s ~ S which leads to an agreement at period 6 t, without too much loss of generality, let U ( s ) =-NBd-2 u(w, ~ , l * ( w ) ) and I I ( s ) = N ~ / - 2 7 r (w, l * ( w ) ) . For any strategy profile s ~ S which leads to perpetual disagreement, let U(s) = 0 and I I ( s ) = O. Let A be the length of the bargaining period. In this paper, we focus on the case where the interval between offers and counteroffers is short, i.e. as the period length A shrinks to zero. One can imagine the delay between successive offers as representing a bargainer's speed of response. With this interpretation of the sequence of offers and counteroffers, a focus on the limit where these delays become arbitrarily short seems compelling. We express the bargainers' discount factors in terms of discount rates, r u and rf, by the formula ~i = e x p ( - r i A), for i = u, f. Greater patience implies a lower discount rate and a higher discount factor: r u >__rf ¢* Bu < ~e- We denote by G ( r u, rf) the wage bargaining game with complete information about the players' discount rates in which the period length A shrinks to zero.

5 W e limit the exposition on pure strategies, but we don't restrict players not to play mixed strategies. All results allow bargainers to play mixed strategies as well as pure ones. 6 The players' payoffs are discounted from period t = 2 since a wage agreement cannot be reached earlier.

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It can be shown that the bargaining game G(r u, rf) possesses a unique limiting subgame perfect equilibrium (SPE). Let (U *(r u, rf), H *(r u, re)) be the unique limiting SPE payoff vector of G ( r u, rf), which is obtained when the length A of a single bargaining period approaches zero. Binmore et al. (1986) have shown that the unique limiting SPE of Rubinstein (1982) alternating-offer bargaining game approximates the Nash bargaining solution to the appropriately defined bargaining problem. Their result can be extended to our wage bargaining game: the unique limiting SPE of G(r u, rf) approximates the asymmetric Nash bargaining solution to our bargaining problem (Y, d), where the parameter a (Union's bargaining power) is computed as follows: rf

(8)

ru+r~

Thus the predicted wage is WcSPE = arg maXw ~ x [ N u ( w, ~)] ~ [ N~r ( w, ~)] 1-~ = arg maXw~ x [ ( a - w ) ( w - ~)] " [ a - w] 2~1 ~)

(9)

This is WcSPE = w + ( a / 2 ) ( a - ~ ) , where a is the Union's bargaining power. In other words, the more impatient the bargainer, the less powerful. Thus the unique SPE payoff vector of G ( r u, re) is (U*(r.,

rf), H * ( r u , re) ) =

(N(2- ~)a 4b(N+l) (a-w)2' U(2

~)2~

- 0~) 2

4b(N+

l) 2

(a

-

)

(10)

However, both the asymmetric Nash bargaining solution and the Rubinstein's model predict efficient outcomes of the bargaining process (in particular agreement is settled immediately). This is not the case once we introduce incomplete information into the wage bargaining game, in which the first rounds of negotiation are used for information transmission between the two players. The main feature of the bargaining game is that players possess private information. They are uncertain about each others' discount rates. It is common knowledge between the players that player i's discount rate is included in the set Jr/v, r/q, where 0 < r/v ~ r~ < 1, for i = u, f. The superscripts T and 'P' identify the most impatient and most patient types, respectively. The players' types are independently drawn, with player i's discount rate drawn from the set [riP, r~] according to the probability distribution Pi, for i = u, f. Letting p = (pu, Pf), we denote by G(p) the wage bargaining game of incomplete information in which the distribution p is common knowledge between the players (and in which the period length A shrinks to zero). Next we state some properties about the perfect

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363

Bayesian equilibria (PBE) of G(p). A pure strategy of player i is a function si:[r f, rti] × H i ~ A . Let /z be a system of beliefs where /z(.lh) = (/zu(.lh), /zf(. [h)) are the beliefs of the players regarding each others' types, conditional on history h. We denote by Uro (17r) the payoff function of the Union (Firm) with discount rate r u (re). A PBE in the game G(p) is a (s, /z) that satisfies: (i) sequential rationality: for each h ~ H, the continuation strategies are a Bayesian equilibrium in the continuation game given the beliefs /x(. [h); (ii) correct initial beliefs: /x(-I•) = p; (iii) player i's belief about his opponent does not change as a result of player i's own actions; (iv) Bayes' rule is used to update beliefs whenever possible. The following lemma follows from Watson (1994) analysis.

Lemma 1. Consider the wage bargaining game G( p). For any PBE of G( p): - Union's payoff belongs to [ U * ( r t, rf), U *(rue, r])]; Firm's payoffbelongs to [17 *(re, r]), 17 *(rtu, rf)]. Remember that (U *(r u, rf), H * ( r u, rf)) is the unique limiting SPE payoff vector of G(r u, rf). For solving this wage bargaining game (with incomplete information), we could use a weaker solution concept, like Iterative elimination of strictly conditionally dominated strategies 7 (IECDS) for games with incomplete information, instead of the equilibrium approach (PBE). Mutual knowledge that both players will play a strategy that survives IECDS leads to bounds on the payoffs which may arise in the game; bounds which are the same as in Lemma 1 (see Vannetelbosch, 1995 or Watson, 1994). Lemma 1 is not a direct corollary to Watson (1994) Theorem 3 because Watson's work focuses on linear preferences, but the analysis can be modified to handle the present case. As Watson (1994) stated, Lemma 1 establishes that "each player will be no worse than he would be in equilibrium if it were common knowledge that he were his least patient type and the opponent were his most patient type. Furthermore, each player will be no better than he would be in equilibrium with the roles reversed". The next proposition follows from Lemma 1.

Proposition 1. The wage bargaining outcome at the industry level, w~, satisfies the following inequalities:

4 ~+2(r~+rP)(a-~)
).

(11)

7 Fudenberg and Tirole (1991) were first to define the procedure of 1ECDS and the notion of

conditional dominance but for games with complete information. Watson (1994) adapted the procedure of IECDS to games with incomplete information. Watson (1994) has also characterized the set of strategies that survive IECDS and shown how IECDS restricts the behaviour of the players in Rubinstein's alternating-offerbargaining game with two-sided incomplete information.

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In the alternating-offer bargaining game G(p) with incomplete information, PBE implies bounds on the centralized wage outcome, Wc*,which are given by Eq. (11). The lower (upper) bound is the wage outcome of the complete information game, when it is common knowledge that the Union's type is r ui (r~) and the Firm's type is rfP (rI). This lower (upper) bound is a decreasing function of Union's discount rate r ui (r~), an increasing function of Firm's discount rate rfP (r]), an increasing function of the level of industry demand, parameterized by the intercept of the linear demand function, and an increasing function of the reservation wage N. Lemma 1 and Proposition 1 tell us that inefficient outcomes are possible, even as the period length shrinks to zero. The wage bargaining game may involve delay, but not perpetual disagreement, at equilibrium. Indeed, Watson (1994) has constructed a bound on delay in equilibrium which shows that an agreement is reached in finite time and that delay time equals zero as incomplete information vanishes. Eq. (11) implies bounds on the firm's employment level Ic (i.e. firm's output) at equilibrium. 2r I + rp 2( rol+rfP)

(a-

N)

b(N+l) >l;>

2r~e + r] 2(rP+r[)

(a-

N)

b(N+l)'

(12)

The lower (upper) bound is an increasing function of Union's discount rate r~ (ruP), a decreasing function of Firm's discount rate rfP (r[), an increasing function of the level of industry demand, and a decreasing function of the reservation wage

4. Decentralized wage bargaining Now, we consider a situation of decentralized wage bargaining. Inside each firm, the union whose objective function is to maximize his utility and the management (or the firm) whose objective function is to maximize his profits negotiate the wage level. All negotiations take place simultaneously and independently. When negotiating the wage level, each union-firm pair takes all other wage settlements in the industry as given. Moreover, union-firm pairs always correctly anticipate their effects on the Cournot competition in the product market. The bargaining process describing the negotiation between union and firm inside each firm 8 is the same as the one developed in Section 3. Let (Un* ( r u, r f ) , -/]n* ( r u , r e ) ) be the unique limiting SPE payoff vector at firm n of the decentralized wage bargaining game under complete information, which is

8 Remember that all unions have the same discount factor 6 u and all firms have also the same discount factor 6t.

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V.J. V a n n e t e l b o s c h / E u r o p e a n J o u r n a l o f Political E c o n o m y 13 (1997) 3 5 3 - 3 7 4

obtained when the length A of a single bargaining period approaches zero. Therefore, in complete information, the decentralized wages are given by W l SVE d ~

argmax[Ul(W ' , ~ , / l , ( W 1. . . .

, WN))]a[.g.l(Wl

,

WN))] ~ "

l ; ( w 1. . . •.

wI~X

Wnd"SPE =

[ u(,

argmax

( *w I , . . . , W . . . . .

w.,~,l~,

.))]°

' WN

w~EX X

sPE_argmax --

WNd

[ (

"1"1" n Wn

, l n* ( W 1* .

[ u N ( WN, ~ ,

(.

IN W I

.,w N ))11_o

,w.,

. . . . .

, •

.. , WN

))]o[ ~rN ( w N I N. ( ,w I . . . . ,

.

wN

))]1-o

wNEX

which can be rewritten as

wSd°E = argmax

[a--Nwl+W:, b-~-+ 6

(w,- ~)

¢-b~N + 1-)

Wl~X

a -

.SPE

W n o = argmax

Nw n + W * -n

b( N + 1)

wnEX

]~[ a - N w . + W_~.

(Wn-')l [

[a--NwN+W-*N

,SPE

wNd =

arg m a x WNE X

-b-(N+-l-)

_ ( WN

I'~Ia-NWN+W*-N] W)

]2(1

-~)

]

2(1-~)

C b ( U + 1)

(13) where ~ is the local union's bargaining power and it is given by Eq. (8). There is a unique solution to Eq. (13) given by wdSPE

SPE = Wld

• ..

. SPE = = WNd

~_F

ot( a - ~ )

2N- a(N-

=~+ 11

rf( a - ~ ) 2Nr u + (N+

1)rf'

Therefore, the SPE payoff vector (at firm n) is

N(2-~)(a-~)

2

(N(2-~))

(Un* ( ru, rf), Hn* ( ru, rf) ) = b( N - - ~ T ) ~ 2 N ~-~-( N ~ I) ] 2 o r , - ~ _ ~ )

.

Next we tackle the decentralized wage bargaining game with incomplete informa-

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tion about the players' discount rates. Given the symmetry of the model, we look for symmetric PBE, that is, an equilibrium in which Wla = w2*~ = . . . = WNO = Wd .

Lemma 2. Assume each union-firm pair n takes all other wage settlements in the industry as given during the bargaining at firm n. Then, f o r any symmetric PBE: 1 . - union n's ( o f type ru) payoffbelongs to [Un* (r u,1 rfP ), Un* (r uP , rj;)], * P r I n* 1 firm n's ( o f type rf) payoff belongs to [1I, (ru, f ) , 17 (ru, r jP) ] ,. f o r r i E [ri P, ri 1], i = u, f. Lemma 2 is the counterpart of L e m m a 1 for the decentralized wage negotiation. Following L e m m a 2 and the complete information results we are able to state some properties about the decentralized wage outcomes.

Proposition 2. The (symmetric) wage bargaining outcome at the firm-level, w~ , satisfies the following inequalities: rf~'( a - ~ )

r~( a - ~ )

+ 2Nr~ + ( N + 1)rfP < wa* < ~ + 2 N Pu + ( N + 1)r~"

(14)

In Eq. (14), the lower (upper) bound is the wage outcome of the complete information game, when it is common knowledge that union's type is r ui ( r ~ ) and firm's type is r~' (rI). This lower (upper) bound is a decreasing function of union's discount rate r ui (ruP), an increasing function of firm's discount rate r~ (r~!), an increasing function of the level of industry demand, and an increasing function of the reservation wage ~. Note that, even as A approaches zero, potential inefficiency is possible in the presence of incomplete information. Eq. (14) implies bounds on firm's employment level ld* at equilibrium.

2 Nr~ + Nr~ 2NrXu + ( N + l ) r ~

(a-F) b(N+

[ 1) > l ~ > [ 2 N Pu + ( N +

(a-W) l)r~

b(N+

1)

(15)

In this model of wage determination in an oligopolistic industry, the wage level is lower under decentralized bargaining than under centralized bargaining when there is complete information 9. Is this result always true when bargainers possess

9 When wage bargaining takes place at the firm-level, each union-finn pair expects to be able to alter its relative wage position in the industry. Therefore, it results wage spillover effects: each union-finn pair has an incentive to lower wages in order to increase its output level (or employment level) and the finn's profits, incentive which increases with the number of firms in the industry. When wage bargaining takes place at the industry-level, these wage spillover effects vanish and wage bargaining affects employment and profits only through the overall level of industry demand.

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private information? Comparing Eq. (11) with Eq. (14), we obtain the following result: Result 1. In an unionized oligopolistic industry, if wage bargainers possess private information, then the decentralized wage outcome will not necessarily be lower than the centralized wage outcome. The following inequality (Eq. (16)) is the necessary and sufficient condition such that the wage outcome is always strictly lower under decentralized bargaining than under centralized bargaining. 1-N r~ r u' + - - r 2

) P
(16)

Whether or not Eq. (16) is satisfied depends on two factors: the incomplete information or uncertainty about the players' discount rates and the number of finns in the industry. Incomplete information in the model takes into account two main features. The first one is the amount of private information in possession of the players. By the amount of private information we mean the size of the set in which player's discount factor is contained and which is common knowledge between the players. The second one is the uncertainty about who is the more patient player (i.e. who is the stronger player). When it is common knowledge that the union is stronger, this second feature disappears, and information tends to play a less crucial role in the process of the negotiation between finns and unions. Therefore, it is more likely to recover the complete information's result where the industry-level wage outcome is always larger than the firm-level wage outcome. The number N of finns producing in the industry has only an effect on the decentralized wage outcome: wa* depends negatively on N, while Wc* does not depend on N. Therefore the decentralized wage outcome is more likely to be lower than the centralized wage outcome the bigger is N. The negative relationship between N and wd may be explained by the following argument. As already mentioned, if each union-finn pair n expects to be able to alter its relative wage position in the industrial sector, then union-finn pairs have some incentive to cut wages in order to gain a larger share of the product market. This incentive increases with the number N of firms operating in the industrial sector. When N becomes large, the decentralized (symmetric) wage outcome will be closer to the reservation wage (i.e. the perfectly competitive wage outcome). In the limit for N ~ ~, the decentralized (symmetric) wage equals the reservation wage even when there is incomplete information. Therefore, it is not surprising that a sufficient condition which satisfies Eq. (16) is r P ~< r ui < r f ~< r~ and N > 3. This sufficient condition means that it is common

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knowledge among players that the union is stronger (more patient) than the firm, while discount factors remain private information, and the number of firms in the industry is greater than two.

Proposition 3.

If r e ~ r I < r[ <~rJ and N >_ 3, then %* > w 2 .

4.1. Lopsided convergence The previous analysis establishes bounds on the PBE payoffs, but it says nothing about the possible payoff vectors inside the bounds. Remark that Watson (1994) has also studied the PBE payoff set of Rubinstein's alternating-offer game under various distributions of the types of players, with the support of the distribution held fixed 10. His main result and analysis can be extended to our wage bargaining game: " a slight chance of being a patient type can't help a player, whereas a slight chance of being impatient can certainly hurt". For example, suppose that there are three possible types for both the unions and the firms: r P, ri*, r~ where r S < ri* < rli, for i = u, f. Suppose the distribution over these types (r S, ri*, r I) is (fl, 1 - 2/3, /3) for both the unions and the firms;/3 is the probability that player i's discount rate is r P, 1 - 2/3 is the probability that player i's discount rate is ri*, .... Then, we might wish to know how the set of PBE payoffs or the wage outcomes change as /3 converges to zero, where there is only a slight chance that player i is either of type r P or type r I. From Watson (1994) Theorem 5, it follows that, as /3 converges to zero, PBE wage outcomes are such that: ~+

~+

r f*

2(ri+rr, )(a-F)_
<~+

~) 2 N r l u + ( N + 1)re*
2(r *

(a-N)

r~(a--~) 2Nr u + ( N +

l)re t"

This result tells us that the PBE wages do not converge to a single wage, despite that the distribution over types converges to a point mass distribution. Moreover, this convergence result is lopsided (see Watson, 1994). Indeed, suppose that the firm's type is known, while the union's type is private information to him: the union's discount rate is either rue or r u with rue < r u. I Then, we have that: (i) if the probability of the patient type (rue) is small then the PBE wages will be close to the unique SPE wage of the complete information game in which the union's discount rate is rI; (ii) if the probability of the impatient type (r~) is small then there is a wide range of PBE wages; for the centralized case, there are PBE in

m If r~ and r/~ converge (for i = u, f) then the PBE payoffs of the incomplete information game converge to the unique SPE payoff vector of some complete information game.

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369

which the wage is close to ~ + [rf/(2(rPu + r e ) ) ] ( a - ~ ) and there are PBE in which the wage is close to ~+[rf/(2(rlu + r f ) ) ] ( a - ~ ) . For this one-sided incomplete information game, the necessary and sufficient condition such that the wage outcome is always strictly lower under decentralized bargaining than under centralized bargaining becomes: 2 - I Nr u* ]. r~> N - 1 [ru -

(17)

This condition will be satisfied if, for example, the firm is sufficiently impatient. If there is a slight chance that the union is patient then Eq. (17) always holds. However, if there is a slight chance of being impatient then Eq. (17) might be violated 11

5. Strike activity and private information W e turn to the investigation of the relationship between the bargaining structure and the strike activity. In the wage bargaining model developed in the paper, the union strike decision is taken as exogenous. Therefore, delay in reaching an agreement coincides with the length of a strike. W e investigate the potential inefficiency and delay (which is bounded at equilibrium) implied by a centralized and a decentralized bargaining system for the case when the length between two offers tends to zero. W e construct an indicator for the scope of potential inefficiency as the difference between the upper bound and lower bound on the wage outcome 12. W e denote it ~ . When bargaining takes place at the industry level, the indicator of potential inefficiency, qsc, is: rfi ,r u - -

4r u ~ c = 2 ( r ~ + rl)( r~ + rfP) ( a - - F ) .

(18)

W h e n bargaining takes place at the firm level, the indicator of potential inefficiency, Wd, is: 2 N ( r fI r uI - - r fP r uP )

atta = (2Nr~ + ( N + 1)rlf)(2Nr~ + ( N + 1 ) r ~ ) ( a - ~ ) .

(19)

l i This lopsided convergence follows from the construction of PBE strategies. Watson (1994) stated: "... players will punish one another if they depart from their equilibrium strategies. An effective form of punishment in the bargaining game is that when a player takes some deviant action, beliefs about him are updated optimistically-putting probability one on his weakest type... The existence of a very impatient type (a type near r~ as compared to ri* ) allows the threat of such a revision of beliefs, however small is the probability of the impatient type. The existence of a very patient type has little effect, since it would not be used in punishing a player". 12The larger is the difference between the upper bound and the lower bound on the wage outcome, the larger is the potential inefficiency incurred during the negotiation.

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and ~d are increasing (decreasing) functions of r uI (r~) and are dercreasing (increasing) functions of rfP (rfI). When we compare Eq. (18) with Eq. (19) we observe that aPc > ~ . We can state the following result.

Result 2. Potential inefficiency is larger if wage bargaining takes place at the industry level than at the firm level. This result is supported by Cheung and Davidson (1991) in their two-period model 13 and by evidence from the Belgian labour market. Indeed, from Table 1 (source: Vannetelbosch, 1996), it is clear that since the late seventies and beginning eighties, the negotiations have become more and more decentralized and the strike activity (i.e. the number of strikes having for motive wages) has decreased a lot (see also Vannetelbosch (1996) for an empirical investigation of wage determination in Belgium). These facts seem to corroborate the model developed in the paper as well as an incomplete information framework for investigating wage negotiations. We turn to the explanation that strikes are more likely to outcome and have a longer duration when bargaining takes place at the industry-level. For example, consider a market consisting of two identical unionized firms. At firm 1, for any given w 2 there will be a Pareto frontier representing the maximum profit firm 1 can earn for any given level of utility achieved by the union. The aim of the bargaining between the firm and the union is to agree on a point (hence, a wage and an agreement's time) in a set which is bounded above by this Pareto frontier. Notice that higher values of w 2 increase firm l ' s competitiveness and shifts out the Pareto frontier. Therefore the two Pareto frontiers which define the upper bounds on the range of possible payoffs of the wage bargaining game are interdependent. When the unions bargain at the firm-level they ignore this interdependence, while when agents on the same side merge and form an industrywide coalition they internalize this interdependence. That is, this interdependence or wage spillover effects are internalized partially by collective bargaining at the industry-level, which raises the potential payoffs but, in expanding the payoff set (or range of possible payoffs), also increases the scope for delay and inefficiency. Finally, note that when bargaining takes place at the firm-level, the potential inefficiency depends negatively on the number N of firms producing in the market. As N ~ o,, the decentralized wage outcome equals the reservation wage

~3 Remember that Cheung and Davidson (1991) have investigated the relationship between the bargaining structure and the strike activity in a two-period model with incomplete information, where the strike decision is endogenous. They have found that incentives are created that lead to a greater expected level of strike activity when a single bargaining agent represents the interests of all workers. Cheung and Davidson (1991) have also reported that coalition bargaining on the employers' side had led to an increase in strike activity in Canada.

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and is unique. Therefore as N becomes large, potential inefficiency tends to disappear. This last result is not valid under centralized bargaining because we* does not depend on N.

6. Conclusion In this paper, we have developed a model of wage determination with private information in an unionized oligopolistic industry. We have compared the outcome of collective bargaining under two different bargaining structures - - one in which the wage is bargained at the industry-level and one in which the wages are bargained at the firm-level. We have described the bargaining process by Rubinstein (1982) alternating-offer bargaining model with two-sided incomplete information about the bargainers' discount rates. Iterative elimination of strictly conditionally dominated strategies (see Watson, 1994) for games with incomplete information leads to bounds on the payoffs which may arise in the game and to bounds on the wage agreements that may be made. Moreover, the perfect Bayesian equilibria payoffs and wages conform to these bounds. In a private information framework, main results are: firm-level (or decentralized) wage outcome is not necessarily lower than industry-level (centralized) wage outcome and potential inefficiency (hence, possible strike activity) is larger when bargaining takes place at the industry level. The first result suggests that the impact of the negotiation level on the economy is not robust to a change in the information structure. Thereby economic policies supported by results within a complete information framework have to be taken cautiously. Some extensions may be worthwhile. A first one is to extend the wage bargaining model to incorporate the choice of calling a strike, so that the union strike decision is endogenous. A second one is to endogenize the choice of the bargaining structure; an attempt has been made by Petrakis and Vlassis (1996) for the complete information framework. A third one is to investigate whether the results on the relationship between the bargaining structure and the wage determination are robust once we consider a more flexible time structure (for example, the production stages and the negotiation periods could be staggered).

Acknowledgements I would like to thank Pierre Dehez for helpful discussions and suggestions. I also wish to acknowledge the comments of two anonymous referees.

Appendix A. On dominance in Cournot game Does iterative elimination of strictly dominated strategies (IEDS) solve the Cournot game C(N, ~ , ~.N)? First, we give the normal-form definition of strict

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dominance for the Cournot game C(N, ~ N , ,D.N). Notice that if IEDS yields a unique strategy profile then this strategy profile is necessarily a Nash equilibrium and it is the unique Nash equilibrium of the game (Fudenberg and Tirole, 1991).

Definition 1. 1 Given a product set T = 1-[~~ N Tn ~ ~ N of strategy profiles, a player n ~ N, and a pair q., q'n ~ T. of firm n's strategies, strategy q. strictly dominates q'. relative to T if rr.(q., q_ .) > 7r.(q'., q_ .) for all combinations q - n E T n ~" I-ImE N\{n}Tm o f all the other firms' strategies. This definition tells us that a strategy of a firm n is strictly dominated if there is another strategy of firm n that gives firm n strictly higher profit independent of what all other firms do. Let D , ( T ) denote the members of T, that are not strictly dominated by other members of Tn relative to T. Let D ( T ) = l-l,~ NDn(T). For each k ~ N 0, let D k =- D ( D k- 1) be defined recursively, starting from D O= ~vD,k denotes firm n's component of the product space D k = 1-[~ ND~; it is the set of all n's strategies that remain after k rounds of eliminating all the strictly dominated strategies of every firm. Evidently, O ¢ D,k _c D~-I_C_ ... c Dn~ ___D ° = R + ( k = 3, 4 . . . . ).

Definition 2. The game C(N, ~N+, ,.B.N) is solvable by iterated strict dominance oz ~¢ k if, for each firm n ~ N, D n =- • k D~ is a singleton. Proposition 4. Given parameters a, b and wages ( w , ) ~ N, define N 1 as the lowest integer such that f~(QM_n) = 0 Vn, n = 1. . . . . N 1. Then, (a) f o r N = 2, D~ = {q~*(wj, w2)}, n = l, 2; (b) f o r N > _ N , , D~ = [0, qM], n = 1. . . . . N; (c) f o r N ~ ( 2 , N1), 0 v~D~___[O, qM], n = 1. . . . . g. Proof. D ° = ~ + V n ~ N. Notice that Vn ~ N:ern(q,, q_ n) is strictly concave in q~ and f ~ ( Q - n ) is continuous and downward sloping. Let qff denote the monopoly outputs: qff =f~(0) for n = l . . . . . N. The first round of elimination of strictly dominated strategies yields Dn1 = [0, qff] for n = 1. . . . . N. Indeed, producing outputs strictly greater than the monopoly output are strictly dominated since a firm cannot obtain a higher profit than the monopoly profit. The second round of elimination of strictly dominated strategies yields D,,2 [fn(QM_n), qM]. Three cases may occur: (a) V n ~ N:f~(QM_~)> 0, (b) V n E N:fn(QM_n) 0, (C) 3n, m ~ N:fn(QM,,) > 0 and fm(QMm) = 0. The occurrence of one of these cases depends on the number N of firms in the industry. Since f,(QM_,) is decreasing with N for all firms, for N = 2 we have case (a), for N > N 1 > 2 we have case (b), and for N ~ (2, N l) we have case (c). Case (a) has been shown by Fudenberg and Tirole (1991, pp. 47-48). Assume case (b). Then it follows immediately that D~ = [0, q ~ ] = D 3 = ... = D~ Vn ~ N. Assume case (c). Then we have: D~ ___ ... C D 2 _ D ~ = [ O , q M ] V n ~ N . [] =

=

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Proposition 4 tells us the following. For N = 2, IEDS solves the game. Therefore assuming a Nash equilibrium in the Coumot duopoly game C ( N = 2, ~ N , zrN) doesn't seem too demanding. For N > 3, IEDS does not solve the game. Indeed, for N > N 1, any output between zero and the monopoly output qff survives IEDS. For N ~ (2, N1), it may be that the monopoly output qnM does not survive IEDS. If we want to overcome the problem of an undetermined outcome with IEDS for the Cournot game C ( N > 2, ~+u, 7rN), then we have to assume some coordination of firms' behaviour such that the firms produce the unique Nash equilibrium output of the game. Corollary 1. If w, = w Vn(n = 1. . . . . N), then: (a) for N = 2, D~ = {q,* (w, w)}, n = 1, 2; (b) forN>_3, D~ = [ 0 , qff], n = 1. . . . . N. Remark that Basu (1992) has characterized the class of rationalizable equilibria of quantity-setting oligopoly games wherein the firms are identical in the sense of having identical cost function. It is shown that when the industry market is small, the rationalizable solution is not different from the Nash equilibrium outcome. But as the size of the industry market increases the set of rationalizable outcomes grows and converges to the interval from zero to the monopoly output. By a large industry market, Basu (1992) means that, for a given and fixed demand, the number of firms becomes large. Note that Corollary wage-dominance coincides with the characterization of the rationalizable solution in the special case where the demand and cost functions are linear (see Basu, 1992 or Bernheim, 1984). BiSrgers and Janssen (1995) have considered an alternative definition of large markets. Their definition requires that not only the number of firms but also the number of consumers becomes large. That is, Btirgers and Janssen (1995) have considered the Cournot game in which not only the supply side of the industry market, but also the demand side is replicated. Then, provided that the number of replications of the industry market exceeds some finite boundary, there will be a unique outcome of the Cournot game that survives the iterative elimination of strictly dominated strategies if and only if the equilibrium of the approximated perfectly competitive industry market is globally stable under cobweb dynamics.

References Binmore, K.G., Rubinstein, A., Wolinsky, A., 1986. The Nash bargaining solution in economic modelling. Rand Journal of Economics 17, 176-188. Basu, K., 1992. A characterization of the class of rationalizable equilibria of oligopoly games. Economics Letters 40, 187-191. Bernheim, D., 1984. Rationalizable strategic behavior. Econometrica52, 1007-1028. B~Argers,T., Janssen, M.C.W., 1995. On the dominance solvability of large Cournot games. Games and Economic Behavior 8, 297-321. Cheung, F.K., Davidson, C., 1991. Bargaining structure and strike activity. Canadian Journal of Economics 91,345-371.

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Davidson, C., 1988. Multiunit bargaining in oligopolistic industries. Journal of Labor Economics 6, 397-422. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press. Horn, H., Wolinsky, A., 1988. Bilateral monopolies and incentives for merger. Rand Journal of Economics 19, 408-419. Petrakis, E., Vlassis, M., 1996. Endogenous Wage-Bargaining Institutions in Oligopolistic Industries, Discussion Paper 96-02. Universidad Carlos III de Madrid, Spain, Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97-109. Vannetelbosch, V.J., 1995. On Dominance in Wage Bargaining with Incomplete Information in an Unionized Cournot Oligopoly, Discussion Paper 95-03. University of Louvain, Louvain-la-Neuve, Belgium. Vannetelbosch, V.J., 1996. Testing between alternative wage-employment bargaining models using Belgian aggregate data. Labour Economics 3, 43-64. Watson, J., 1994. Dominance and Equilibrium in Alternating-Offer Bargaining, Discussion Paper 94-13. University of California, San Diego.