Strategic union delegation and strike activity

Transcription

1 Strategi union delegation and strike ativity Ana Mauleon FNRS and CEREC, Faulte s Universitaires Saint-Louis Vinent Vannetelbosh FNRS and CORE, Universite atholique de Louvain Abstrat. We develop a model o wage determination with private inormation, in whih the union has the option to delegate the wage bargaining to either surplus-maximizing delegates or to wage-maximizing delegates (suh as senior union members). We show that the wage outome in ase o surplus-maximizing delegates is not neessarily smaller than the wage outome in ase o wage-maximizing delegates, even when the wage bargaining with private inormation is lose to one with omplete inormation. However, i it is ommonly known that the union is stronger than the irm and the demand is suiiently elasti, then delegating to wage-maximizing delegates deinitely inreases the wage at equilibrium. The maximum delay in reahing an agreement is greater whenever the union hooses wage-maximizing delegates instead o surplus-maximizing delegates and remains inite even when the period length shrinks to zero. JEL lassiiation: C70, C71, C72, C78 De le gation syndiale strate gique et ativite de gre`ve. Nous étudions omment l option pour les syndiats de de léguer la ne goiation salariale va inluener le salaire négoié et les initations a` aire la gre` ve. Nous de veloppons un modèle de négoiation salariale ave inormation inomplète dans lequel le syndiat a l option de déléguer la ne goiation à un délégué qui maximise le surplus ou a` un de légué qui maximise le salaire. Nous montrons que le salaire négoié par un délégue qui maximise le surplus n est pas ne essairement inérieur au salaire négoié par un délégue qui maximise le salaire.cependant,si le Ana Mauleon and Vinent Vannetelbosh are Cherheurs Qualiiés at the Fonds National de la Reherhe Sientiigue, Belgium. We wish to thank two anonymous reerees or valuable omments. The researh o Ana Mauleon has been made possible by a ellowship o the Fonds Européen du Développement Eonomique Re gional (FEDER). Finanial support rom the Belgian Frenh Community s program Ation de Reherhes Conertée 99/ and rom the researh projet BEC (Spanish government) is grateully aknowledged. vannetelbosh@ore.ul.a.be Canadian Journal o Eonomis / Revue anadienne d Eonomique, Vol. 38, No. 1 February / évrier Printed in Canada / Imprimé au Canada / 05 / / # Canadian Eonomis Assoiation

2 150 A. Mauleon and V. Vannetelbosh syndiat est plus ort que la irme et la demande est élastique, alors le ait de déléguer la ne goiation à un délégue qui maximise le salaire va augmenter le salaire d équilibre. Finalement, nous montrons que de plus longues gre` ves sont observe es lorsque le syndiat de lègue la ne goiation à un délégue qui maximise uniquement le salaire. 1. Introdution Our purpose in this paper is to provide a theoretial study o how the option or unions to delegate the wage bargaining will aet the wage outome and the inentives or strike ativity. Up to now the literature has been mainly oused on strategi delegation on behal o shareholders. Fershtman and Judd (1987) have addressed the issue o strategi managerial delegation in the ontext o oligopolisti industries with Cournot ompetition (see also Sklivas 1987). Regarding strategi union delegation, Jones (1989a) has shown that a divergene between the objetives o union leaders and union members will naturally arise in a demorati union as part o a rational bargaining strategy. Essentially, the reason is that in many bargaining situations, ommitment an be valuable, and the union members an redibly ommit to a bargaining stane, whih they ould not otherwise sustain, by delegating authority to a negotiator whose objetives make this stane an optimal one. In ase the union and the irm bargain over wages and employment levels, union delegation implies that ineiient bargains are reahed (on a pseudo ontrat urve), whih are preerred by the union to the eiient bargaining solution (on the true ontrat urve). In ase the union and the irm only bargain over the wage (and the irm sets employment unilaterally), union delegation may mimi the monopoly union outome where the union hooses its most preerred wage. More reently, Conlin and Furusawa (2000) have provided an explanation o why senior union members may represent the union in ontrat negotiations with a monopolist when parties also negotiate about the bargaining agenda. By strategially delegating ontrat negotiations to wage-maximizing individuals, the surplus-maximizing union may be better o than i surplus-maximizing individuals negotiate the ontrat. But these previous studies have onsidered omplete inormation rameworks so that delay in reahing an agreement (strikes or lokouts), whih waste industry resoures, annot our at equilibrium. 1 So, we go beyond the analysis oered in Jones (1989a) and Conlin and Furusawa (2000) by developing a model that enable us to investigate in presene o strategi union delegation how private inormation aets the wage level and the delay in reahing an agreement. 2 1 Strikes data seem to have a signiiant impat on the wage-employment relationship or olletive negotiations (see, e.g., Kennan and Wilson 1989). 2 See Kennan and Wilson (1989, 1993) or surveys o bargaining models with private inormation and their relation to strike data. See Kennan (1986) or a survey o the empirial results on strike ativity.

3 Strategi union delegation and strike ativity 151 Preisely, we develop a model o wage determination in whih both the union and the irm have private inormation. In the irst stage, the union hooses whether to use surplus-maximizing delegates or to use wage-maximizing delegates (suh as senior union members or workers who are proteted rom being dismissed) who will negotiate the wage with the employer. So, only the union has the option to delegate bargaining authority. As pointed out by Jones (1989a), it is the dierential ability to ommit (here, the union an delegate bargaining authority to workers who annot be laid o), not dierenes in the desire to do so, that underlies one-sided delegation models. I one onsiders objetives or negotiators, whih do not represent the true preerenes o any worker, then it may be more appropriate to allow both parties to be able to delegate bargaining authority to a negotiator. 3 In the seond stage, the wage bargaining ours. To desribe the wage-bargaining proess, we adopt Rubinstein s (1982) alternatingoer bargaining model with two-sided inomplete inormation, whih allows the ourrene o strikes at equilibrium. 4 In the third and inal stage, the irm hooses the output level to be produed. As a benhmark we irst onsider the omplete inormation situation, and we show that the weaker the union is, the more likely it is the union will hoose to send wage-maximizing delegates. The hoie o wage-maximizing delegates always inreases the wage level and dereases the prodution output (and the employment level) as well as the onsumer surplus. One the negotiators have private inormation, the omplete inormation results are not always valid. The wage outome in ase o surplus-maximizing delegates an be greater than the wage outome in ase o wage-maximizing delegates, even when the wage bargaining with private inormation is lose to one with omplete inormation. The less elasti the labour demand is, the more likely it ould happen. However, i it is ommonly known that the union is stronger than the irm and the labour demand is suiiently elasti, then we reover the omplete inormation result, namely, that the wage outome with surplus-maximizing delegates is always stritly smaller than the wage outome with wage-maximizing delegates. The maximum delay in reahing an agreement (or maximum strike ativity) is greater whenever the union hooses wage-maximizing delegates instead o surplus-maximizing delegates. From onventional wage-bargaining theory, irm s monopoly power inreases the surplus to be divided between the union 3 Jones (1989b) has extended Jone s (1989a) analysis o one-sided delegation to symmetri, twosided delegation ase. The negotiators have the same struture o preerenes as the onerned parties, but with potentially dierent taste parameters. A large hoie o negotiators is available to eah party. Although one party may gain at the expense o the other by the proess o delegation relative to sel-representation, it is oten the ase that two-sided delegation leads to utility losses or both parties. 4 The union and the irm bargain only over wages (then, the irm hooses the employment level). This assumption allows us to model the wage-bargaining proess by means o Rubinstein s (1982) alternating-oer bargaining model, and to use Watson s (1998) analysis o Rubinstein s model in the ase o two-sided inomplete inormation.

4 152 A. Mauleon and V. Vannetelbosh and the irm. Monopoly power (prie-ost margin) is greater when the irm aes wage-maximizing delegates. As a result, delegating the wage bargain to wage-maximizing delegates raises the potential payos or the union and the irm but, in expanding the payo set (or range o possible payos), also inreases the sope or delay (longer strikes and lokouts may be needed or sreening the private inormation). We show by means o an example that strategi union delegation an inrease substantially the potential delay. Finally, we provide the neessary and suiient ondition suh that, even in presene o private inormation, it is always optimal or the union to hoose wage-maximizing delegates. The paper is organized as ollows. In setion 2 the model is presented. In setion 3 we desribe the wage-bargaining game and hoie o delegates under omplete inormation. Setion 4 is devoted to the wage bargaining with private inormation. It oers some preditions regarding the strike duration. Finally, setion 5 onludes. 2. The basi model Consider a market or a single homogenous produt, where the demand is given by P ¼ a b Q, P is the market prie, Q is the quantity produed, and > 0. There is one irm produing the good. Let denote the proit level. The only variable input is labour. Tehnology exhibits onstant returns to sale and is normalized in suh a way that Q ¼ L, where L is labour input, and the unit prodution ost o eah irm is the wage W. Thus, the proit o eah irm is given by ¼ (a bq )Q WQ: (1) Without loss o generality, we let b ¼ 1. The irm belongs to and is ontrolled by one risk-neutral owner, whose objetive is to maximize proits. In addition, the irm is unionized, and enters into a losed-shop agreement with its riskneutral union. The union objetive is to maximize the union surplus: U ¼ L(W W), (2) where W is the reservation wage. The wage rate is determined by negotiations between the irm and the union delegates. Preeding the negotiations, the union may aet the negotiation outome by seleting delegates whose objetive is either to maximize the union s surplus or to maximize the wage rate. We develop a three-stage game. In stage one, the surplus-maximizing union hooses whether to use surplus-maximizing delegates or to use wage-maximizing delegates (suh as senior union members) who will negotiate the wage with the employer. The objetive o a wage-maximizing delegate is simply V ¼ W W. In stage two, the wage bargaining ours. Finally, in stage three the employer hooses the output level. The model is solved bakwards.

5 Strategi union delegation and strike ativity 153 In the last stage o the game, knowing that the wage level (W) has already been determined, the employer hooses Q(W) ¼ a W 1 1 þ to maximize its proits. In stage two, the negotiation takes plae. We irst onsider the omplete inormation bargaining as a benhmark. 3. Wage bargaining with omplete inormation First, we onsider the ase in whih the union sends surplus-maximizing delegates whose interest is the same as the union s objetive. The negotiation proeeds as in Rubinstein s (1982) alternating-oer bargaining model. The irm and the union delegate make alternatively wage oers, with the irm making oers in odd-numbered periods and the union delegate making oers in even-numbered periods. The length o eah period is D. The negotiation starts in period 0 and ends when one o the negotiators aepts an oer. No limit is plaed on the time that may be expended in bargaining and perpetual disagreement is a possible outome. The union is assumed to be on strike in every period until an agreement is reahed. Both the irm and the union are assumed to be impatient. The irm and the union delegate have time preerenes with onstant disount rates r > 0 and r u > 0, respetively. 5 To apture the notion that the time it takes to ome to terms is small relative to the length o the ontrat, we assume that the time between periods is very small. This allows a study o the limiting situations in whih the bargaining proedure is essentially symmetri and the potential osts o delaying agreement by one period an be regarded as negligible. As the interval between oers and ounter-oers is short and shrinks to zero, the alternating-oer model has a unique limiting subgame peret equilibrium, whih approximates the Nash bargaining solution to the bargaining problem (see Binmore, Rubinstein, and Wolinsky 1986). Thus, the predited wage is given by Ws SPE ¼ arg max U U 0 0 1, (4) (3) 5 Two versions o Rubinstein s (1982) alternating-oer bargaining model apture dierent motives that indue parties to reah an agreement rather than to insist indeinitely on inompatible demands. In a irst version the parties inentive to agree lies in the at that they are impatient: player i is indierent between reeiving xexp ( r i ) today and x tomorrow, where r i > 0 is player i s disount rate. In a seond version the parties are not impatient, but they ae a risk that i agreement is delayed, then the opportunity they hope to exploit jointly may be lost: player i believes that at the end o eah bargaining period there is a positive probability 1 exp ( r i ) that the proess will break down, r i > 0. Thus, r i an be interpreted either as player i s disount rate or as his estimate about the probability o a breakdown o the negotiations.

6 154 A. Mauleon and V. Vannetelbosh where the lowersript s means that wage bargaining is between the irm and surplus-maximizing union delegates, and where U 0 ¼ 0and 0 ¼ 0are,respetively, the disagreement payos o the union delegate and the irm. The parameter 2 (0,1) is the union bargaining power whih is equal to r /(r u þ r ). Simple omputation gives us Ws SPE ¼ W þ 1 þ (a W) ¼ W þ r (a W): (5) (r u þ r ) (1 þ ) Obviously, the wage is inreasing with the reservation wage W, with the union bargaining power, and with the parameter. 6 Then, one an easily obtain the equilibrium employment level 1 L * s ¼ 1 þ, (1 þ ) 2 (a W) (6) as well as the union s payo and the irm s proit, whih are denoted U * s () and * s (), and are given by U * s () ¼ 1 1 þ (1 ) 1 þ (1 þ ) 2 a W 1þ (7) 1þ * s () ¼ 1 þ (1 ) (1 þ ) 2 (a W) : (8) Seond, we onsider the ase in whih the union sends wage-maximizing delegates. Then, the predited wage is given by Ww SPE ¼ arg max V V 0 0 1, (9) where the lowersript w means that wage bargaining is between the irm and wage-maximizing union delegates, and where V 0 ¼ 0 and 0 ¼ 0 are, respetively, the disagreement payos o the union delegate and the irm. The parameter 2 (0,1) is still the union bargaining power whih is equal to r / (r u þ r ). Simple omputation gives us W SPE w ¼ W þ 1 þ (a W) ¼ W þ r (a W): (10) (1 þ )r u þ r Again, the wage is inreasing with the reservation wage W, with the union bargaining power, and with the parameter. Then, one an easily obtain the equilibrium employment level 6 The larger is, the less elasti (or the more inelasti) the demand is. Thus, the larger is, the more inlined the irm is to aept high wages, sine the irm an easily pass wages on pries without loosing too many onsumers.

7 Strategi union delegation and strike ativity 155 L * w ¼ 1 (1 þ ) a W 1, (11) as well as the union s payo and the irm s proit, whih are denoted U * w () and * w (), and are given by U * w () ¼ 1 þ 1 (1 ) a W (1 þ ) * w () ¼ ð1 Þ (1 þ ) a W 1þ 1þ (12) : (13) From (5), (8), (10), and (13) we obviously get that Ww SPE > Ws SPE and * s () > * w (). A natural question to ask at this point is whether union delegation redues onsumer surplus or soial welare. We denote by CS the onsumer surplus. It is equal to CS s ¼ 1þ 1 þ 1 þ (1 þ ) 2 (a W) (14) or the ase in whih the union sends surplus-maximizing delegates, and it is equal to CS w ¼ 1þ 1 (a W) (15) 1 þ 1 þ or the ase in whih the union sends wage-maximizing delegates. Comparing both expressions yields that the onsumer surplus is always lower when the union sends wage-maximizing delegates rather than surplus-maximizing delegates. In the irst stage o the game, the union hooses whether to use surplusmaximizing delegates or wage-maximizing delegates to negotiate the wage with the employer. Comparing (7) with (12) we obtain the ollowing proposition. PROPOSITION 1. The union will send wage-maximizing delegates i and only i (1 þ ) þ2 (1 ) (1 þ ) þ1 (1 þ ): Proposition 1 tells us that (i) or any given union bargaining power 2 (0,1), the more inelasti the produt demand is (i.e., is big), the more likely the union will hoose to send wage-maximizing delegates; (ii) or any given degree o elastiity o the demand, the weaker the union is (i.e., is small), the

8 156 A. Mauleon and V. Vannetelbosh more likely the union will hoose to send wage-maximizing delegates. In ase the produt demand is linear, ¼ 1, the union will hoose to send wagemaximizing delegates i and only i *^.76. So, i the union is relatively not too strong, then the union will delegate the negotiation task to wagemaximizing delegates. 7 Proposition 1 gives us the optimal hoie in terms o delegation under omplete inormation, subjet to the restrition that the union an only hoose between surplus-maximizing or wage-maximizing delegates. In ase one allows or a broad lass o objetives or union delegates or negotiators, Jones (1989a) has shown that appropriate seletion o a negotiator an mimi the monopoly union outome where the union selets its most preerred wage (on the labour demand urve). For instane, i the union delegates to a negotiator with objetive L (W W) 1þ and gives appropriate inentives or the negotiator, 8 the outome o the negotiation will be the monopoly union wage. But one suh objetive or the negotiator (an objetive whih does not represent the true preerenes o any worker) is onsidered, it may be more appropriate to allow both parties, the union and the irm, to be able to delegate bargaining authority to a negotiator. Both the asymmetri Nash bargaining solution and the Rubinstein s model predit eiient outomes o the bargaining proess (in partiular agreement is reahed immediately). This is not the ase one we introdue inomplete inormation into the wage bargaining, in whih the irst rounds o negotiation are used or inormation transmission between the two negotiators. 4. Wage bargaining with private inormation 4.1. Peret Bayesian equilibria The main eature o the negotiation is that both negotiators have private inormation. Eah negotiator does not know the impatiene (or disount rate) o the other party. It is ommon knowledge that the irm s disount rate is inluded in the set [r P, ri ], where 0 < rp r I, and that the union s disount rate is inluded in the set [r P u, ri u ], where 0 < rp u ri u. The supersripts I and P identiy the most impatient and most patient types, respetively. The types are independently drawn rom the set [r P i, ri i ] aording to the probability distribution p i, or i ¼ u,. We allow or general distributions over disount rates. This unertainty implies bounds on the union bargaining power whih are denoted by ¼ r P ri u þ 1 rp and ¼ r I r P u þ 1. ri 7 As pointed out by Jones (1989a), one has to be autious with respet to the interpretation o eonometri estimates o trade unions objetives done in the past, sine these estimates did not distinguish between the objetive o the trade union and the objetive o the union delegate who atually negotiated. See Penavel (1991) or a survey o the empirial results on trade union objetives. 8 Jones (1989a) has designed a monetary payment shedule that provides appropriate inentives at all levels o ees and hene allows us to set the ee to zero.

9 Strategi union delegation and strike ativity 157 LEMMA 1. Consider the wage bargaining with inomplete inormation in whih the distributions p and p u are ommon knowledge, and in whih the period length shrinks to zero. For any peret Bayesian equilibria (PBE), the payo o the union belongs to U * (), U * () and the payo o the irm belongs to [ * (), * ()]. The proo o this lemma, as well as the other proos, may be ound in appendix A. In lemma 1, U * () and * () denote, respetively, the SPE utility o the union and the SPE proit o the irm o the omplete inormation game, when it is ommon knowledge that the union s bargaining power is ¼. In ase the union hooses to send surplus-maximizing delegates, then U * () and * () are given, respetively, by expression (7) and expression (8) with ¼.In ase the union hooses to send wage-maximizing delegates, then U * () and * () are given, respetively, by expression (12) and expression (13) with ¼. Similarly or ¼. Lemma 1 ollows rom Watson s (1998) analysis o Rubinstein s alternating-oer bargaining model with two-sided inomplete inormation. 9 Lemma 1 is not a diret orollary to Watson s (1998) theorem 1, beause Watson s work ouses on linear preerenes, but the analysis an be modiied to handle the present ase. Translating Watson s (1998) theorem 2 to our ramework ompletes the haraterization o the PBE payos. LEMMA 2. Consider the wage bargaining with inomplete inormation in whih the period length shrinks to zero. For any eu 2 [U * (), U * ()], e 2 [ * (), * ()], there exists distributions p u and p, and a PBE suh that the PBE payos are eu and e. In other words, whether or not all payos within the intervals given in lemma 1 are possible depends on the distributions over types. As Watson (1998) stated, lemma 1 and lemma 2 establish that eah player will be no worse than he would be in equilibrium i it were ommon knowledge that he were his least patient type and the opponent were his most patient type. Furthermore, eah player will be no better than he would be in equilibrium with the roles reversed. Sine we allow or general distributions over types, multipliity o PBE is not an exeption (even when the game is almost with omplete inormation; see setion 4.2). From lemma 1 we obtain proposition 2 and proposition 3. PROPOSITION 2. For any distribution over types, PBE wage bargaining outomes in ase o the union hooses to send surplus-maximizing delegates, W * s (, ), satisy the ollowing inequalities: 9 Watson (1998) haraterized the set o PBE payos that may arise in Rubinstein s (1982) alternating-oer bargaining game and onstruted bounds (whih are met) on the agreements that may be made. The bounds and the PBE payos set are determined by the range o inomplete inormation and are easy to ompute beause they orrespond to the SPE payos o two bargaining games o omplete inormation. These two games are deined by mathing one player s most impatient type with the opponent s most patient type.

10 158 A. Mauleon and V. Vannetelbosh W þ r P (r I u þ rp ) (1 þ ) (a W) W* s (, ) W þ r I (r P u þ ri ) (a W): (16) (1 þ ) Notie that eah wage satisying these bounds an be the outome by hoosing appropriately the distribution over types. The lower (upper) bound is the wage outome o the omplete inormation game, when it is ommon knowledge that the union s type is r I u (rp u )andtheirm stypeisrp (ri ) (and the union bargaining power is ()). For many distributions over types, thereisawiderangeopbe:therearepbeinwhihthewageisloseto the upper bound, and there are PBE in whih the wage is lose to the lower bound (see setion 4.2 or some intuition). Expression (16) implies bounds on the irm s employment level, as well as on the irm s output, at equilibrium. PROPOSITION 3. For any distribution over types, PBE wage bargaining outomes in ase o the union hooses to send wage-maximizing delegates, W * w (, ), satisy the ollowing inequalities r P W þ (1 þ )r I u þ rp (a W) W * w (, ) W þ r I (1 þ ) r P u þ (a W): (17) ri In omplete inormation, the hoie o wage-maximizing delegates always inreases the wage level and dereases the prodution output (and the employment level) as well as the onsumer surplus. But one the union and the irm have private inormation, these omplete inormation results do not neessarily hold. COROLLARY 1. In ase o union-irm bargaining with private inormation, the wage outome with surplus-maximizing delegates is not neessarily smaller than the wage outome with wage-maximizing delegates. Let ¼ rp ri ri u ri þ rp u rp r I u ri rp u rp ¼ (1 þ ) : (18) be the uto value on whih is obtained by equating the lower bound rom (17) with the upper bound rom (16). The larger the amount o private inormation j j is, the smaller the uto value is. Whenever >, the standard result o omplete inormation may not hold: sending wage-maximizing delegates might derease the wage level in equilibrium. Indeed, when the labour demand is inelasti, the objetive o the surplus-maximizing delegate and the objetive o the wage-maximizing delegate beome lose. Then, one an always ind probability distributions over types suh that there is a PBE wage

11 Strategi union delegation and strike ativity 159 with surplus-maximizing delegates whih is higher than some PBE wage with wage-maximizing delegates. The neessary and suiient ondition suh that the omplete inormation result always holds is. The more elasti the labour demand is, the more likely the wage outome in ase o wagemaximizing delegates will be higher than the wage outome in ase o surplusmaximizing delegates even in presene o inomplete inormation. There are industries in whih unions have a dominant power. One example is the International Typographial Union (ITU) beore the seventies. See Dertouzos and Penavel (1981), Penavel (1991). COROLLARY 2. Suppose it is ommonly known that the union is stronger than the irm ( 2 3 ). For any distribution over types, i the produt demand is suiiently elasti ( 1), then W * w (, ) > W* s (, ). The intuition behind this orollary is the ollowing one. Inomplete inormation in the model takes into aount two main eatures. The irst one is the amount o private inormation in possession o the players. By the amount o private inormation we mean the size o the set in whih player s disount rate is ontained and whih is ommon knowledge between the players. The seond one is the unertainty about who is the more patient player, that is, who is the stronger player. When it is ommon knowledge that the union is stronger than the irm, this seond eature disappears, and inormation tends to play a less ruial role in the proess o the negotiation between the irm and the union. The more elasti the produt demand is, the smaller the mark-up o prie over marginal ost is, and the less unertainty there is in the wage bargain. Moreover, i the elastiity o produt and labour demands is high, a wage inrease will imply a signiiant drop in employment level and, hene, it will rerain surplusmaximizing delegates rom demanding high wages. Thereore, the omplete inormation result is reovered one it is ommon knowledge that the union is stronger than the irm and that the produt demand is suiiently elasti Wage bargaining with almost omplete inormation The previous analysis establishes bounds on the PBE payos, but it says nothing about the possible payo vetors inside the bounds. It would be interesting to study the set o payos and the set o wages that are supported by peret Bayesian equilibria in the wage-bargaining game whih is lose to having omplete inormation. Watson (1998) has also studied the PBE payo set o Rubinstein s alternating-oer game under arbitrary sequenes o distributions over the players types whih have the same (possibly wide) support, 10 yet whih onverge to a point mass distribution. That is, he has examined bargaining games in whih with high probability a player s disount rate is 10 I r P i and r I i onverge (or i ¼ u, ), then the PBE payos o the inomplete inormation game onverge to the unique SPE payo vetor o some omplete inormation game.

12 160 A. Mauleon and V. Vannetelbosh lose to a ertain value, yet there is a slight hane that the player s disount rate is muh higher or muh lower. He has shown that the set o equilibrium payos does not onverge to that o the omplete inormation, despite that the game onverges to one o omplete inormation. More preisely, the set onverges rom above but not rom below in the sense that a player annot gain i there is a slight hane that he is very patient (has a low disount rate), yet he an suer i there is a slight hane that he is impatient. In other words, a slight hane o being a patient type an t help a player, whereas a slight hane o being impatient an ertainly hurt. The limiting set o equilibrium payos is deined by eah player s greatest possible disount rate and the limiting disount rates; the players lowest possible disount rates do not play a role. Watson s main result an be extended to our wage bargaining. It also urnishes intuition that is meaningul or general distributions. Suppose that there is three possible types or both the union and the irm: r P i, r* i, r I i where r P i < r * i < r I i,ori¼u,. Suppose the distribution over these types r P i, r* i, ri i is (, 1 2, ) or both the union and the irm; is the probability that player i s disount rate is r P i,1 2 is the probability that player i s disount rate is r * i,and is the probability that player i s disount rate is ri i. Then, we might wish to know how the set o PBE payos or the wage outomes hange as onverges to zero, where there is only a slight hane that player i is either o type r P i or type r I i. From Watson s (1998) theorem 4 and theorem 5, it ollows that, as onverges to zero, PBE wage outomes satisy the bounds in expression (16) and expression (17) with r P i ¼ r * i. There are PBE in whih the wage is lose to the upper bound W SPE (r * u, ri ) and there are PBE in whih the wage is lose to the lower bound W SPE (r I u, r* ). So, PBE wages do not onverge to a single wage, despite that the distribution over types onverges to a point mass distribution. 11 COROLLARY 3. There exists union-irm bargaining with almost omplete inormation where the wage outome with surplus-maximizing delegates is not neessarily lower than the wage outome with wage-maximizing delegates. Suppose that the irm s type is known, while the union s type is private inormation to him: the union s disount rate is either r P u or ri u with rp u < ri u. Suppose that the distribution over types (r P u, ri u )is(1 ", ") where" is the probability o the impatient type (r I u ); " is small. The above analysis tells us that there is a wide range o PBE wages. It is possible that, in ase o surplus-maximizing delegates, an outome will be reahed lose to the upper bound wage, 11 This lopsided onvergene ollows rom the onstrution o PBE strategies, where players will punish one another i they depart rom their equilibrium strategies. An eetive orm o punishment in the bargaining game is that, when a player takes some deviant ation, belies about him are updated optimistially putting probability one on his weakest type. The existene o a very impatient type (a type near r i I,omparedwithr i * ) allows the threat o suh a revision o belies, however small the probability o the impatient type. The existene o a very patient type has little eet, sine it would not be used in punishing a player.

13 Strategi union delegation and strike ativity 161 r W þ (1 þ )(r P u þ r (a W); ) meanwhile, in ase o wage-maximizing delegates, an outome would be reahed lose to the lower bound wage, r W þ (1 þ ) r P u þ r (a W): Then, i the demand is not too elasti, that is, > r =(r I u rp u ) 1, the wage outome in ase o surplus-maximizing delegates is higher than the one that would be reahed in ase o wage-maximizing delegates Maximum delay in reahing an agreement Ineiient outomes are possible, even as the period length shrinks to zero. While the sope o possible ineiieny is lear rom lemma 1 and lemma 2, what is not so obvious is the potential or delay. The wage bargaining game may involve delay (strikes or lok-outs), but not perpetual disagreement, at equilibrium (see Watson 1998). 12 In at, delay is positively related to the distane between the disount rates o the most and least patient types o the players. I the range o types is redued, then this leads to a smaller range o possible payos and less delay. Delay an our even when the game is lose to one o omplete inormation (as the type distributions onverge to point mass distributions). We propose to identiy strike ativity with the maximum delay time in reahing an agreement. Only in average this measure is a good proxy or atual strike duration. 13 It is not unommon in the literature on bargaining to analyse the maximum number o delay time that may be expended beore reahing an agreement. 14 In the appendix we ompute the maximum delay in equilibrium 12 Watson (1998) has onstruted equilibria with delay, in whih the types o eah player behave identially (no inormation is revealed in equilibrium), players use pure strategies, and players make non-serious oers until some appointed date. 13 In the literature on strikes, three dierent measures o strike ativity are usually proposed: the strike inidene, the strike duration, and the number o work days lost due to work stoppages. See Cheung and Davidson (1991), Kennan and Wilson (1989). Sine we allow or general distributions over types and we may have a multipliity o PBE, we are unable to ompute measures o strike ativity as the ones just mentioned. 14 Cramton (1992) has onstruted a sequential equilibrium in a bargaining model with two-sided unertainty, where types are revealed ater a maximum o two rounds but delay is diretly related to the types o the players. There is also a ontinuum o other sequential equilibria where types are not ully revealed. In ontrast to Rubinstein s model, in whih the time between oers is ixed, in Cramton (1992) eah player an delay making oers. This ability to delay oers enables eah player to ommit to not revising or resinding an oer until a ounter-oer is made. Cai (2003) has also analysed the maximum number o delay periods but in a multilateral bargaining.hismodelhasa inite number o Markov peret equilibria, some o whih exhibit wasteul delays. The maximum number o delay periods that an be supported in Markov-peret equilibria inreases in the order o the square o the number o players. These results are robust to a relaxing o the Markovrequirements and to more general surplus untions. Ausubel, Cramton, and Denekere (2002) provide a reent survey o bargaining models with inomplete inormation.

14 162 A. Mauleon and V. Vannetelbosh whih shows that an agreement is reahed in inite time and that delay time equals zero as inomplete inormation vanishes (in that r P i and r I i onverge). In ase the union hooses surplus-maximizing delegates, the maximum real delay time in reahing an agreement is given by D s ¼ min D u s, D s (19) where 2 D u s ¼ 1 r P log4 u r P u þ ri r I u þ rp 1þ r I u þ 1 1 þ rp r P u þ 1 1 þ ri!1 r P r I 3 5 (20) is the maximum real time the union would spend negotiating, and " D s ¼ 1 r P 1 þ r P u log þ! 1 1 þ ri r I u þ # rp r I u þ 1 þ 1 rp r P u þ ri (21) is the maximum real time the irm would spend negotiating aing a surplusmaximizing union delegate. In at, D u s (D s ) is the maximum real time the union (irm) would spend negotiating in any equilibrium i the union (irm) was o its most patient type. So, D u s (D s ) is the upper bound on the maximum time the union (irm) o type r u (r ) would spend negotiating; maximum time that is dereasing with type r u (r ). So, the more patient a player is the greater the delay that may be observed. Sine D u s and D s are positive, inite numbers, the maximum real delay in reahing an agreement in ase o surplus-maximizing delegates is inite and onverges to zero as r I i and r P i beome lose, or i ¼ u,. In ase the union hooses wage-maximizing delegates, the maximum real delay time in reahing an agreement is given by D w ¼ min D u w, D w, (22) where " D u w ¼ 1 r P u r P log þ! 1 þ ri r P # u r I u þ 1 þ rp is the maximum real time the union would spend negotiating, and " D w ¼ 1 r P 1 þ r I u log þ! 1 þ rp r P # u r P u þ r I u r I 1 þ ri (23) (24) is the maximum real time the irm would spend negotiating aing a wagemaximizing union delegate. Sine D u w and D w are positive, inite numbers, the

15 Strategi union delegation and strike ativity 163 TABLE 1 Maximum delay in reahing an agreement ¼ ½ ¼ 1 ¼ 2 r I D s D w D s D w D s D w (366) (43) maximum real delay in reahing an agreement in ase o wage-maximizing delegates is inite and onverges to zero as r I i and r P i beome lose, or i ¼ u,. Comparing (20) with (23) and (21) with (24), we have D u w > Du s and D w > D s or rp i < r I i, i ¼ u,. It ollows that D w > D s. PROPOSITION 4. The maximum delay in reahing an agreement is greater whenever the union hooses wage-maximizing delegates instead o surplus-maximizing delegates. That is, D w > D s or r P i < r I i. From onventional wage bargaining theory, irm s monopoly power inreases the surplus to be divided between the union and the irm. Monopoly power (prie-ost margin) is greater when the irm aes wage-maximizing delegates. 15 As a result, delegating the wage bargain to wage-maximizing delegates raises the potential payos or the union and the irm but, in expanding the payo set (or range o possible payos), also inreases the sope or delay (longer strikes and lokouts may be needed or sreening the private inormation). We provide now an example about the maximum delay. In this example we let r P ¼ r P u ¼ rp, r I ¼ ri u ¼ ri and r P ¼ 0.22 r I with r I 2 [0.11, 0.21]. Table 1 gives us the integer part o the maximum delay or dierent values o the parameter. 16 We observe that (i) D w > D s or r P i < r I i,(ii)d w and D s 15 In omplete inormation, wages are higher in ase o wage-maximizing delegates, but pries are muh higher too. Indeed, surplus-maximizing delegates are about employment (hene, output), whih pushes pries down. Moreover, high wages tend to push pries up. 16 We an interpret r i as the annual disount rate and the numbers in table 1 as the maximum number o days needed to reah an agreement. Indeed, the integer part o the maximum delays or ¼ 1/365 are exatly the numbers o Table 1, exept when there are brakets (then it is the number in brakets). The data in Table 1 seem onsistent with U.S. strike durations reported in Cramton and Tray (1994).

16 164 A. Mauleon and V. Vannetelbosh are inreasing with the amount o private inormation jr P i r I i j, (iii) D w is dereasing with and D s is inreasing with (so the dierene between D w and D s is beoming larger the more elasti the demand is). Notie that real delay time in reahing an agreement is not negligible: many bargaining rounds may be needed at equilibrium beore an agreement is reahed. Results (i) and (ii) hold in general, while result (iii) does not. However, i the union has more private inormation than the irm has, it is likely that the dierene between D w and D s is beoming larger the more elasti the demand is. Indeed, we u w =@ < s =@ > 0 i r I u rp u > 1 1þ r I rp. From proposition 4 we know that i the union hooses to send wagemaximizing delegates then strike ativity is going to inrease. Beore onluding, we briely investigate whether and when it is optimal to delegate or the union. The neessary and suiient ondition suh that it is always optimal or the union to hoose wage-maximizing delegates is U * w () ¼ (1 ) 1 þ (1 ) 1 þ (1 þ ) 1 þ (1 þ ) 2 ¼ U * s (), (25) where U * w () and U* s () are given by expressions (12) and (7), respetively. Take the ase o a linear demand ( is equal to 1). Then, the above ondition (25) beomes (1 ) (2 ) : 2 (2 ) 8 For instane, i it is ommonly known that the union is weaker than the irm (i.e., 1 2 ) and the union is not too weak (i.e., 2 5 ) then it is optimal or the union to send wage-maximizing delegates. 5. Conlusion We have developed a model o wage determination with private inormation, in whih the union has the option to delegate the wage bargaining to either surplus-maximizing delegates or to wage-maximizing delegates (suh as senior union members). We have shown that the wage outome in ase o surplusmaximizing delegates is not neessarily smaller than the wage outome in ase o wage-maximizing delegates, even when the wage bargaining with private inormation is lose to one with omplete inormation. However, i it is ommonly known that one party is stronger than the other and the labour demand is quite elasti, then delegating to wage-maximizing delegates inreases or sure the wage at equilibrium. We have shown that strike ativity is greater whenever the union hooses wage-maximizing delegates instead o surplusmaximizing delegates. We have also ound that strategi delegation an (26)

17 Strategi union delegation and strike ativity 165 inrease substantially delays. From a poliy perspetive our analysis questions whether one should allow or strategi union delegation (e.g., by means o laws proteting union delegates rom being dismissed). From a researh perspetive our analysis questions theoretial results obtained under omplete inormation as well as empirial studies o the trade union objetives. Appendix A: Proos The proo o lemma 1 is done in two main steps. First, we will show that all strategies that survive the iterative elimination o onditionally dominated strategies impose some bounds on the players payos (see lemma A1). Seond, we will show that eah PBE onorms to lemma A1. There are two players who must agree on a wage W rom the set X, where X is the set o easible agreements: X {W 2 R j 0 W a}. The players either reah an agreement in the set X or ail to reah an agreement, in whih ase the disagreement event E ours. The two players have well-deined preerenes over X [ E. The negotiation proeeds as in Rubinstein s (1982) alternating-oer bargaining model. The irm and the union delegate alternatively make wage oers, with the irm making oers in odd-numbered periods and the union delegate making oers in even-numbered periods. The negotiation starts in period 0 and ends when one o the negotiators aepts an oer. No limit is plaed on the time that may be expended in bargaining, and perpetual disagreement is a possible outome. The union is assumed to be on strike in every period until an agreement is reahed. Both the irm and the union are assumed to be impatient. The irm and the union (delegate) have time preerenes with onstant disount ators 2 (0, 1) and u 2 (0, 1), respetively. Let H i be the set o histories ater whih player i has the move, or i ¼ u,. Let H H u [ H. Let h ¼ 0/ be the history at the start o play. A pure strategy o player i is some untion s i :H i! A that maps eah possible history ater whih player i has the move into an ation. In eah period, the ation spae or the player who makes an oer is X, while the ation spae or the responder is {aept, rejet}. Let S i be the set o strategies or player i, i ¼ u,. Let S S u S. Eah strategy proile s 2 S indues an outome whih speiies agreement at some date or irrevoable disagreement. Payos in the wage bargaining are given as untions o the players strategy proile aording to the vn-m utility untions Y i :S! R. For any strategy proile s 2 S that leads to an agreement W at period n, let Y (s) ¼ n (W, L(W)) and Y u (s) ¼ u n U(W, L(W)) be, respetively, the irm s payo and the union s payo. In ase the union hooses a wage-maximizing delegate, the payo o the delegate is simply u n V(W). For any strategy proile s 2 S that leads to perpetual disagreement, let Y i (s) ¼ 0. For h 2 H i, let Y i (s j h) be the payo o player i in the game onditional on h desribing the play and s desribing the play thereater. Deine Y i i as the payo untion o player i with disount

18 166 A. Mauleon and V. Vannetelbosh ator i. The wage-bargaining game with omplete inormation about the players disount ators has a unique subgame peret equilibrium (SPE). The SPE wage outome in ase o surplus-maximizing delegates is suh that (W u, L(W u )) ¼ (W, L(W )) U(W, L(W )) ¼ u U(W u, L(W u )), where W u is the SPE wage outome i the union delegate makes the irst wage oer, and W is the SPE wage outome i the irm makes the irst oer. Sine the union makes the irst oer, the SPE wage is W SPE s ( u, ) ¼ W þ 1 ( ) 1 þ 1 u (a W), rom whih we get the SPE proit and the SPE union s payo: U * s ( u, ) ¼ 1 ( ) 1 þ * s ( u, ) ¼ 1 1 þ ( ) 1 þ 1 þ u 1 þ 1 a W 1 u ( ) 1 þ u (a W) 1 u 1þ 1 þ (A1) : (A2) Similarly, the SPE wage outome in ase o wage-maximizing delegates is suh that (W u, L(W u )) ¼ (W, L(W )), V(W ) ¼ u V(W u ) where W u is the SPE wage outome i the union delegate makes the irst wage oer, and W is the SPE wage outome i the irm makes the irst oer. Sine the union delegate makes the irst oer, the SPE wage is W SPE w ( u, ) ¼ W þ 1 ( 1 þ ) (a W), 1 u ( ) 1 þ rom whih we get the SPE proit and the SPE union s payo: V * w ( u, ) ¼ 1 ( 1 þ ) (a W) (A3) 1 u ( ) 1 þ * w ( u, ) ¼ 1 1 þ 1 þ ( ) 1 þ (1 u ) (a W) 1 u ( ) 1 þ 1 þ : (A4) The players have private inormation. They are unertain about eah others disount ators. Player i s disount ator is inluded in the set

19 Strategi union delegation and strike ativity 167 i I, P i, where 0 < I i i P < 1, or i ¼ u,. The supersripts I and P identiy the most impatient and most patient types, respetively. For solving the wage-bargaining game with inomplete inormation about the players disount ators, we use weaker solution onepts or games with inomplete inormation. DEFINITION 1. Take a set o strategy proiles T ¼ T u T Ì S suh that T is a produt set, and a type i o player i. Strategy s i is stritly onditionally dominated or type i with respet to T i there exists some s 0 i 2 S i suh that Y i i (si 0, s jjh) Y i i (s i, s j jh) or all s j 2 T j and all h 2 H, and there exists h 0 2 H or whih Y i i (si 0, s jjh 0 ) > Y i i (s i, s j jh 0 ) or all s j 2 T j. A strategy s i is stritly onditionally dominated i there exists a strategy s 0 i suh that s 0 i perorms at least as well as s i ater all histories and stritly better ater at least one history, subjet to s j 2 T j. DEFINITION 2. Take a set o strategy proiles T ¼ T u T S suh that T is a produt set, and a type i o player i. Strategy s i is weakly onditionally dominated or type i with respet to T i there exists some si 0 2 S i suh that Y i i (si 0, s jjh) Y i i (s i, s j j h) or all s j 2 T j and all h 2 H, and there exists sj 0 2 T j and h 0 2 H or whih Y i i (si 0, s j 0jh0 ) > Y i i (s i, sj 0jh0 ). DEFINITION 3. A strategy s i is stritly (weakly) onditionally dominated with respet to T Ì S i and only i s i is stritly (weakly) onditionally dominated or eah type i 2 [ I i, P i ]. Let SD i (T) be the set o strategies or player i that are not stritly onditionally dominated with respet to T, andletsd(t) SD u (T) SD (T). For eah positive integer k 1, let SD k SD(SD k 1 ) be deined reursively, starting rom SD 0 S. {SD k ; k 0} is a weakly dereasing sequene, i.e. 0/ 6¼ SD k þ 1 SD k, k ¼ 1, 2, The limit set is given by SD 1 lim k!1 SD k ¼ T 1 k ¼ 0 SDk.ThesetSD 1 ontains the strategy proiles that remain ater the iterative elimination o stritly onditionally dominated strategies. Let WD i (T) bethesetostrategiesorplayeri that are not weakly onditionally dominated with respet to T, and let WD(T) WD u (T) WD (T). We deine another iterative proess or deleting dominated strategies. The proess involves the reursive deinition that starts rom (WD(SD 1 )) 0 S and goes on with (WD(SD 1 )) k (WD(SD 1 (WD(SD 1 )) k 1 )) or eah positive integer k 1. (WD(SD 1 )) k anbewrittenasthecartesianprodut (WD(SD 1 )) k u (WD(SD1 )) k. {(WD (SD1 )) k ; k 0} is a weakly dereasing sequene; that is, 0/ 6¼ (WD(SD 1 )) kþ 1 (WD(SD 1 )) k, k ¼ 1, 2,...,1. The limit set is given by UD lim k!1 ðwdðsd 1 ÞÞ k ¼ T 1 k ¼ 0ðWDðSD1ÞÞ k. This limit set is the set o players strategy proiles that are undominated by surviving the autious iterative elimination o onditionally dominated strategies.

20 168 A. Mauleon and V. Vannetelbosh Lemma A1 haraterizes the set UD. Let Y * u (I, P u )(Y* u (P, I u )) be the union s unique SPE payo in the game o omplete inormation with disount ators u P (I u )andi (P ), where the union makes the irst wage oer. For instane, in ase o surplus-maximizing delegates Y * u (I, P u )isgivenby expression (27) with ¼ I and u ¼ u P. Let Y* (P, I u )(Y* (I, P u )) be the irm s unique SPE payo in the game o omplete inormation with disount ators u I (P u )andp ( I ), where the irm makes the irst wage oer. LEMMA A1. All s 2 U D are suh that the union s payo Y u (s) 2 U * (u I, P ), U * (u P, I ) and the irm s payo Y (s) 2 * ( I, P u ), * ( P, I u ), or i 2 [i I, P i ], i ¼ u,. Proo. First, we haraterize the set SD 1. The method used is similar to that employed by Shaked and Sutton (1984) and others (e.g., Vannetelbosh 1999). This method looks or bounds on the behaviour o the players and observe that a bound or one player implies a relative bound or the opponent. In turn, this implies a new bound on the behaviour o the original player. For instane, i the union knows that the irm never expets more than (W 0 ) in a period where the irm makes an oer, then the union an guarantee itsel a payo lose to U(W 1 ) by oering a wage W 1 suh that P(W 0) ¼ (W 1 ); obviously, W 1 > W 0. Iterative appliation o this logi leads to the haraterization. Watson (1994) did it or the ase o linear preerenes. Sine the omplete inormation game has a unique SPE, this logi leads to well-deined bounds. Strategy proiles that survive the iterative elimination o stritly onditionally dominated strategies, s 2 SD 1, are suh that (i) the union never oers the irm a wage suh that irm s payo (undisounted) exeeds PY* (I u, P ), (ii) the irm never oers the union a wage suh that union s payo (undisounted) exeeds u PY* u (I, P u ), (iii) the union always aepts any wage that gives him a payo (undisounted) more than u PY* u (P u, I ) and rejets all wages that give him a payo (undisounted) less than u I Y* u (I u, P ), (iv) the irm always aepts any wage that gives him a payo (undisounted) more than PY* (P, I u ) and rejets all wages that give him a payo (undisounted) less than IY* (I, P u ), (v) the union o type u always aepts any wage that gives him a payo (undisounted) more than u Y * u (P u, I ) and rejets all wages that give him a payo (undisounted) less than u Y * u (I u, P ), (vi) the irm o type always aepts any wage that gives him a payo (undisounted) more than Y * (P, I u ) and rejets all wages that give him a payo (undisounted) less than Y * (I, P u ). Seond, we haraterize the set UD. Notie that perpetual disagreement is even possible with strategies belonging to SD 1. For example, the union (o type u ) may rejet a wage giving him a payo (undisounted) o u Y * u (P u, I ) expeting that his ounter-oer wage giving him a payo (undisounted) o Y * u (P u, I ) will be aepted by the irm in the next period. A similar reasoning an be made or the irm. Thereore, bargainers may even play a strategy