B argain
in
gw ith E nd ogen
ousDead l
ines
¤
Ana M auleon
LAB O R E S,
U niversit¶e c atholique d e Lille,
and
IR E S,
U niversit¶
e c atholique d e Louvain.
Vincent Vannetelb osch
yF NR S and IR E S,
U niversit¶e c atholique d e Louvain.
August 2 0 0 1
A bstract
W e develop a two-person negotiation model with complete information which makes endogenous both the deadline and the levelofsurplus destruction afterthe deadline.W e showthatthe equilibrium outcome is always unique butmightbe in-e±cient.M oreover,as thebargainingperiod becomes shortoras theplayers become verypatient,theuniqueoutcomeis always ine±cient.
JEL Classi¯cation:C78,J50 ,J52.
Keywords:bargaining,alternating-o®ers,deadlines,completeinformation.
¤Vincent Vannetelb osch isCharg¶e d e R ec herc hesat the FondsNationald e la R ec herc he Sc ienti¯que, B russels.T he researc h ofAna M auleonhasb eenm ad e possib le b y a fellow ship ofthe Spanish governm ent. F inancialsupport from the B elgianFrench Com munity'sprogram Ac tiond e R ec herc hesConcert¶ee9 9 /0 4 -2 35 (IR E S,U niversit¶e c atholique d e Louvain) isalso gratefully ac know led ged .
yCorrespondingauthor ad d ress: IR E S,U niversit¶e c atholique d e Louvain,P lac e M ontesquieu3,B -134 8 Louvain-la-Neuve, B elgium . E -m ail: vannetelb [email protected] l.ac .be T el: 0 0 32 10 4 74 14 2 , Fax: 0 0 32 10 -4 739 -4 5.
1
In
trod uc tion
N egotiations often take place under the pressure of a deadline. T he deadline may be exogenouslyimposed,oroneofthepartiestothenegotiationmayhavechosenthedeadline and made a credible commitmentto it.R ecentwork has focused on bargaining models with exogenous deadlines afterwhich there is no surplus to be divided. See Fershtman and Seidmann [3], M a and M anove [7]. A key feature ofourpaperis thatwe provide a ¯rst attempt to endogenize the deadline in negotiation models. W e considera more generalde¯nition ofadeadline.Itis apointin timeafterwhich thesurplus tobeshared is permanently reduced.
M oreover, we allow the players to choose the level ofsurplus destruction afterthe deadline. So, in case an agreementis notreached before the deadline, the value ofthe underlyingrelationship willbe permanently reduced and thelevelofsurplus destruction willdepend on theactions oftheplayers.T hen,thesurplus availabletotheplayers once an agreementis reached mayactuallybelowerthan itis atthebeginning.
O ne example ofbargaining in the face ofsuch endogenous deadlines are wage nego-tiations between unions and employers. O n the one hand, both parties may have the opportunitytosettleadeadlineafterwhich starts eitherastrikeoralockout.T hedead-linemaybesubjecttolabourlaws :itis common thatastrikeoralockoutrequires few days'notice in ordertobe legal.O n the otherhand,acon° ictmay reduce permanently thepro¯tabilityoftherelationship itselfbya®ecting,forexample,thefuturedemand for the¯rm's products.Indeed,customers maydecidetobuyfrom nowon tosomecompeti-tor. So, in orderto avoid the lostofcustomers, the ¯rm may decide moving (partially orentirely)theproduction toanotherplantorusingreplacementworkers.B utnotonly the ¯rm has actions atits disposalthatdirectly in° uence the pro¯tability ofthe future relationship.Forexample,the union may jeopardized production equipmentby thelack ofscheduled maintenanceorskilled operators.1
In this paper, we develop a two-stage negotiation modelwith complete information between a ¯rm and a union. In the ¯rst stage, the deadline in force during the wage bargaining is chosen. T hat is, the ¯rm and the union choose, respectively, a lockout date(and theintensity ofthelockout)and astrikedate(and theintensityofthestrike). N evertheless,weallowbothpartiestochoosenodeadline.Inthesecondstage,bothparties
1Cutc her et al.[1]have exam ined for the U .S.pressure tac tic sused b yunionsand em ployersto in°uence the proc essinc ollec tive b argainingand itsoutc om es.Inthe past,the threatsofa strike and the im m inent c ontrac t expirationd ead line have b eenc entralfeaturesm otivatingthe partiesto reac h agreem ents.B ut in rec ent years,the ob servationssuggest that m anagem ent threatsregard ingreplac em ent w orkersand plant c losingsor m ovingsare now also a key part ofthe c ollec tive b argainingland scape.
arebargainingoverthedivision ofasurplus which is timedependent.Indeed,beforethe deadlinewewillhaveapeacefulbargaining,wherein each period untilanewagreement is reached both parties continue toproduce and the value added is shared followingthe old wage contract.A fterthe deadline we willhave an open-con° ictbargaining(a strike ora lockouthas occurred), where in each period untila newagreementis reached both parties getnothingand the value added in laterperiods (once an agreementis reached) willbe a®ected by the intensity ofthe con° ictoccurred.T he wage bargainingproceeds followingR ubinstein's [9]alternating-o®erbargainingprocedurewiththe¯rm makingthe ¯rsto®er.
W eshowthattheequilibrium outcomeofournegotiation modelis always uniquebut mightbeine±cient.T heconditiontogetine±ciencyis satis¯edwhenevertheoldwageis relativelysmall,eachplayerhas athis disposalbothactions thatreducesubstantiallythe valueadded in thefutureand actions thathaveonly aminorimpacton thefuturevalue added,and theplayers arenotimpatient.W hich is theintuitionbehind theresult? B oth players would like thatthe otherplayeris the lastmoveratthe deadline and preferably facingthethreatofacon° ictofastrongintensity.A strongcon° ictsimplymeansthatafter thedeadlinethevalueaddedwillbereducedsubstantially.T hen,inordertoavoidhaving toacceptaverylowwageo®erfacingthethreatofaseverelockout,wherethe¯rm would grab mostofthe surplus, itbecomes optimalforthe union to go on strike immediately and to destroy part ofthe future value added, but not too much. So, at equilibrium we observe both players competingtobe the one whowillmake an o®erjustbefore the deadline,butalsotryingtoavoidtohavetomoveatthedeadlinefacingthethreatofavery strongcon° ict.T his can lead toonepartylaunchingthecon° ictimmediatelyand tothe conclusion ofa P areto-dominated agreement.Finally, as the bargainingperiod becomes shortoras theplayers becomeverypatient,theuniqueoutcomeis always ine±cient.
O urtwo-stagenegotiationmodelis relatedtopapersthatderivebargainingine±ciency undercomplete information2, e.g. van D amme, Selten and W inter[1 1 ], Fernandez and
G lazer[2],H allerand H olden [4].O neimportantdi®erenceis thatine±ciency and delay can arisein theseothermodels becausethereexistmultiplee±cientequilibria,whilethe equilibrium is always uniquebutsometimes ine±cientin ourmodel.
A notherstrand ofrelated literature includes papers on bargaining models with ex-ogenous deadlines. In fact, any ¯nite horizon bargaining model can be interpreted as such.Fershtman and Seidmann [3]study acompleteinformation bargainingmodelwith arandom proposerandanexogenous deadlinebeyondwhichthereis nosurplus todivide. T hey assume thata playercannotacceptalowershare ofthe surplus than she has
viouslyrejected duringthebargainingsession.T his endogenous commitmentassumption togetherwith the deadline imply that, forpatientplayers, there is a unique equilibrium where agreements are delayed untilthe deadline.T his resultdepends on the interaction betweentheexistenceofadeadlineandendogenous commitment.A bsenttheendogenous commitment,theotherassumption cannotexplain delay.M aand M anove[7]constructa bargainingmodelwith completeinformation,whoseuniqueequilibrium is such thatearly in the game o®ers are postponed and late in the game agreements are reached orthe deadlineis missed with positiveprobability.T oobtain such equilibrium twoassumptions are introduced to the ¯nite-horizon alternating-o®erbargaining model.T he ¯rstone is strategicdelay.A n alternating-o®ermodelincorporates strategicdelayifaplayeris per-mitted topostponetheimplementation ofhermovewithoutlosingherturn.T hesecond assumption is imperfect player control over the timing of o®ers during the bargaining session.O ®ers and counter-o®ers areexchanged with exogenous random delay.
Inthesepapers justmentioned,thedeadlineis exogenouslydetermined.H ere,weshow thatoncethedeadlineis endogenous,nootherassumptions suchas e.g.endogenous com-mitmentorstrategicdelayis neededtogetauniqueandine±cientequilibrium.M oreover, ourresultmay alsojustify theexistence ofP areto-inferiorphenomenaotherthan strikes orlockouts,suchas tari®wars,debtmoratoria,break-up ofcease-¯res orwars ingeneral.
T he nextsection presents the basic negotiation modeland some preliminary results. In Section 3 wecharacterizetheequilibrium ofthedeadlinestagegameand weshowthat theequilibrium outcomeis always uniquebutmightbeine±cient.Section 4 concludes.
2
T he NegotiationM od el
W e develop a two-stage negotiation model.T he timingofthis negotiation modelis de-picted in Figure 1 . In the ¯rststage, before the wage bargaining starts attime 0 , the
-r r r r
t= ¡1
c hoic e ofd and° ¯rm o®ersa w age t= 0
uniono®ersa w age t= 1
¯rm o®ersa w age t= 2
strike or loc kout d ate, intensity ofc on°ic t
Stage 1 : d ead line stage j j
j Stage 2 : w age b argaining
unionac c epts or rejec ts ¯rm ac c epts or rejec ts unionac c epts or rejec ts t
¯rm and the union choose simultaneously a lockoutdatedf 2 f0;1;2;:::;d;1 g and its
intensity°f2 f° ;°gandastrikedatedu2 f0;1;2;:::;d;1 gand its intensity°u2 f° ;°g,
respectively.W eallowbothparties tochoosenodeadline,e.g.df=1 simplymeans that
the ¯rm decides tonotchoose alockoutdate.A fterwards, the players arecommitted to the deadline and its intensity they have chosen. A deadline rule thatseems reasonable and ¯ts perfectly with wage negotiations is the followingone.L et(d;°)be the deadline and its intensityin forceduringthewagebargainingstage.Itis given by
(d;°)= 8 > > < > > : (df;°f) ifdf< du; (du;°u) ifdu< df; (du;minf°f;°ug) ifdu=df: (1 )
T hatis, the deadline in force and its intensity (d;°)are determined by the minimum of the deadline choices ofthe players, and in case ofties, ties are broken in favourofthe moreintensecon° ict.T his simpledeadlineruleimplies thatthewagebargainingwilltake placefacingeitherthethreatofalockoutorthethreatofastrike.3Finally,(1;¢)denotes thecasewhereboth parties decidetochoosenorastrikedatenoralockoutdate.4
Inthesecondstageofthenegotiationmodel,thedeadlinedandits intensity° settled are common knowledge and both parties begin tonegotiate.T hereis an in¯nitenumber ofperiods,andineachperiodofnormalproductionthe¯rm has avalueaddedofoneunit ofa good which the ¯rm and the union can divide between them.T he union's share is
W 2[0;1 ],the¯rm's shareis 1¡W .T wobargainingphases aredistinguished.B eforethe deadlinewewillhaveapeacefulbargaining,wherein each period untilanewagreement is reached both parties continue toproduce and the value added is shared followingthe old contract.Initiallythewagelevelin theold contractisW 0 > 0 .A fterthedeadlinewe
willhave an open-con° ictbargaining(a strike oralockouthas occurred), where in each perioduntilanewagreementis reachedbothparties getzeroandthevalueaddedinlater periods (onceanewagreementis reached)willbea®ected bytheintensityofthecon° ict occurred,° (where° willbeequalto° or° with° < ° < 1 ).
3For sim plic ity and for the sake ofpresentationw e have c hosena spec i¯c ationthat exc lud esthe possi-b ility ofhavingsimultaneously a loc kout and a strike.How ever,the resultsw e w illopossi-b tainare,under som e w eak c ond ition, qualitatively rob ust to another spec i¯c ationw here c on°ic tshave anad d itive d estruc tive im pac t onthe future value ad d ed to b e shared .
4W e c analso interpret the c om m itm ent assum ptionto start a c on°ic t at the d ead line interm s of negotiators'reputation.Im agine that no agreem ent hasb eenreac hed and that the uniond ec id esnot to go onstrike at the c hosend ead line.Since the strike d ate ispub lic know led ge (d ue to lab our law s),the union w ould loose m ost ofhisreputationfor the on-goingnegotiationasw ellasfor future ones.Inother w ord s, the resultsw e ob tainare rob ust to the c ase w here the c om m itm ent isrevoc ab le b ut the c ost ofrevoking islarge enough.
Forsimplicity,bothpartiesareassumedtohavelinearutilityfunctions,sotheirpayo®s can berepresented bythediscounted sum offutureshares.Fortheunion this is
U =
1 X t= 0
±t¢ut, (2)
whereut=W 0 ift < dand an agreementhas yettobe reached,ut= 0 ift¸dand no
agreementhas been reached,andut=W fort¸sifan agreementis reached atperiods
onW .T hecommon discountfactoris±2(0;1 ).Forthe¯rm wehavecorrespondingly
V = 1 X t= 0 ±t¢v t, (3)
wherevt= 1 ¡W 0 ift< dand an agreementhas yettobe reached,ut= 0 ift¸dand
noagreementhas beenreached,ut= 1 ¡W fort¸sifanagreementis reached atperiod
s < d,andut=° ¡W fort¸sifan agreementis reached atperiods¸d.
T hebargainingproceeds followingR ubinstein's [9]alternating-o®erbargainingproce-dure.T heplayersareassumedtomakeo®ersalternately,oneo®erperperiod,andwithout loss ofgenerality the¯rm is assumed tomakean o®erin thebeginningofperiod 0 .T he union can then acceptorrejectthis o®er.Ifthe union accepts, the bargaining ends.If the¯rm's o®eris rejected theunion makes anewo®erin thenextperiod,which the¯rm accepts orrejects.Ifthe¯rm accepts,thebargainingends.Iftheunion's o®eris rejected the ¯rm makes a newo®erin the nextperiod, and soon untilan agreementis reached. B oth parties areassumed tohaveperfectinformation in thebargainingstage.
W edenoteB (d;°)thebargainingstage,inotherwords thealternating-o®erbargaining game where itis common knowledge thatthe deadline isdand its intensity is°.A s in R ubinstein, one can showthatthe alternating-o®erbargaininggameB(d;°)possesses a uniquesubgameperfectequilibrium (SP E)andthatanagreementisreachedwithoutdelay atperiod 0 .W henthedeadlineinforceis anoddperiodtheSP E wageW ¤(dodd;°)and
payo®s are: W ¤(dodd;°) = (1 ¡±d)¢W 0 + ± d ¢° 1 + ±, (4) U¤(dodd;°) = 1 ¡± d 1 ¡± ¢W 0 + ±d ¢° 1 ¡±2, (5) V¤(dodd;°) = 1 1 ¡±¡ 1 ¡±d 1 ¡± ¢W 0 ¡ ±d ¢° 1 ¡±2. (6)
W henthedeadlineinforceis an evenperiodtheSP E wageW ¤(deven;°)andpayo®s are:
W ¤(deven;°) = (1 ¡±d)¢W 0 + ± d ¢(1 + ±¡°) 1 + ± , (7) U¤(deven;°) = 1 ¡± d 1 ¡± ¢W 0 + ±d ¢(1 + ±¡°) 1 ¡±2 , (8) V ¤(deven;°) = 1 1 ¡±¡ 1 ¡±d 1 ¡± ¢W 0 ¡ ±d ¢(1 + ±¡°) 1 ¡±2 . (9)
W henthedeadlineinforceis settledatperiod0 ,weenterimmediatelyinanopen-con° ict whicha®ects foreverthevalueaddedtobeshared,andtheSP EwageW ¤(0;°)andpayo®s are: W ¤(0;°)= ±¢° 1 + ±,U¤(0;°)= ±¢° 1 ¡±2,V¤(0;°)= ° 1 ¡±2. (1 0 )
W hen no deadline is settled, the SP E wageW ¤(1;¢)and payo®s are (see H aller and H olden [4]):
W ¤(1;¢)=W 0,U¤(1;¢)= W 0
1 ¡±,V
¤(µ;¢)=1 ¡W 0
1 ¡± . (1 1 )
Comparingtheexpressions hereabove,weobtainthenextlemmawhichgives us some ideas aboutthepreferences oftheplayers overtheintensityofthecon° ictatequilibrium given adeadlined.
L emma1 For any deadlinedodd,W ¤(d;°)< W ¤(d;°), U¤(d;°)< U¤(d;°), and V¤(d;°)> V¤(d;°). For any deadlined6= 0 even,W ¤(d;°)> W ¤(d;°),U¤(d;°)> U¤(d;°),andV¤(d;°)< V¤(d;°).
L emma1 tells us that,theunionprefers thenegotiationfacingthethreatofacon° ict ofaweak(strong)intensity ratherthan facingthethreatofacon° ictofastrong(weak) intensity wheneverthe deadline is odd (even).A con° ictofa strongintensity is simply the case where° = °, and a con° ict of a weak intensity is the case° = °. O n the contrary,the¯rm prefers thenegotiation facingthethreatofacon° ictofastrong(weak) intensityratherthan facingthethreatofacon° ictofaweak(strong)intensitywhenever thedeadlineis odd (even).
3 E n
d ogen
ousDead l
ines
W eturn backtostageoneofthenegotiationmodel,wherethe¯rm andtheunionchoose simultaneously a lockoutdatedf 2 f0;1;2;:::;d;1 gand its intensity°f 2 f° ;°gand a
strikedatedu2 f0;1;2;:::;d;1 ganditsintensity°u2 f° ;°g,respectively.5So,astrategy
forthe¯rm is denoted by (df;°f)and astrategyfortheunion is denoted (du;°u)in the
deadlinestagegame.R ememberthatthedeadlinein forceduringthewagenegotiation is the earliestdate amongthe lockoutdate and the strike date.In case ofa tie we simply assumethatthecon° ictwiththestrongestintensitywillstartatthechosendeadline.T his assumption tobreak ties is quite reasonable in case ofwage negotiations.N evertheless, theresults weobtainarequalitativelyrobusttoamoregeneralassumptionwhereties are broken with high probabilityin favourofthestrongestcon° ict.
Since we allowthe ¯rm tochoose (df = 0;°f)and the union tochoose (du= 0;°u),
possibleine±ciency is notexcluded a-priori.In T able1 werepresent(very partially)the matrixofthedeadlinestagegame,where the ¯rm is therowplayerand theunion is the column player. 0;° 0;° 1;° 2;° 1;¢ 0;° ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) 0;° ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) ° ¡W ¤(0;°) 1;° ° ¡W ¤(0;°) ° ¡W ¤(0;°) 1 ¡W ¤(1;°) 1¡W ¤(1;°) 1¡W ¤(1;°) 2;° ° ¡W ¤(0;°) ° ¡W ¤(0;°) 1 ¡W ¤(1;°) 1¡W ¤(2;°) 1¡W ¤(2;°) 1;¢ ° ¡W ¤(0;°) ° ¡W ¤(0;°) 1 ¡W ¤(1;°) 1¡W ¤(2;°) 1 ¡W 0 T able1 :T hestrategicchoiceofadeadline
In the matrix we only give the per-period SP E payo®forthe ¯rm;the per-period SP E payo® for the union is simply the SP E wage. Forsolving the deadline stage game, a naturalconceptwould be the N ash equilibrium (N E).B ut, this conceptfails to exclude strategy pro¯les that seem implausible such asf(0;°);(0;°)g and, moreover, there is a multiplicity of N E.A s a consequence, we propose to use the trembling-hand perfect equilibrium (T H P E)concept as a device to select plausible outcomes among the N ash ones.
In ordertocharacterize the T H P E ofthe deadline stage game, we willuse twowell-known results (see van D amme [1 0 ]). First, since the deadline stage game is a ¯nite two-playergame, the strategy pro¯le ((df;°f);(du;°u))is a T H P E ifand only ifitis a
N E andneither(df;°f)nor(du;°u)is weaklydominatedwithrespecttof0;1;2;:::;d;1 g.
W esaythataplayer's actionis weaklydominatediftheplayerhas anotheractionatleast
5T he set ofd ead line d atesisassum ed to b e ¯nite (inord er to apply w ell-know nresultson¯nite gam es) and to b e id entic alfor b oth parties. T he upper b ound d 2N is large enough to m ake the analysis interesting.O b viously, a d istinct upper b ound for eac h party w ould not m od ify our analysis.M oreover, the resultsw e ob tainare not m od i¯ed ifw e allow m ore thantw o levelsofintensity ofthe c on°ic t.
as good nomatterwhattheotherplayerdoes and betterforatleastsomeaction ofthe otherplayer.Second,every¯nitegamehas atleastoneT H P E.6
From L emma 1 and the weak dominance concept, it is obvious that (df odd;°)is
weakly dominated by (df odd;°). Indeed, (i)against any (du;°u)such thatdu> df
thestrategy (df odd;°)is doingstrictlybetterthan (df odd;°),(ii)againstany (du;°u)
such thatdu< df the strategy (df odd;°)is doing as wellas (df odd;°), (iii)against
any (du;°u)such thatdu=df the strategy (df odd;°)is doingstrictly betterthan (df
odd;°)if°u=° and is doingequalif°u=°.Similarly,onecan showthat(dfeven;°)is
weaklydominatedby(dfeven;°),(duodd;°)is weaklydominatedby(duodd;°),(du6= 0
even;°)is weaklydominated by(du6= 0 even;°),and (du= 0;°)is weaklydominated by
(du= 0;°).
L emma2 T he strategies(df odd;°),(df even;°),(duodd;°),(du6= 0 even;°)and
(du= 0;°)are allweaklydominated.
T hroughoutthepaperwewillfocus on thecasewheretheplayers aresu±cientlypa-tient(±large),whichcanbereinterpretedasiftheintervalbetweeno®ersandcountero®ers is short.7M oreprecisely,wefocus on thecasewhere± > ° > °.
A ssumption 1 ± > °.
L emma3 T he strategies(df even;°),(duodd;°)are allweakly dominated.
Indeed,if± > ° then (df6= 0 even;°)is weakly dominated by (df¡1;°),(df= 0;°)
is weakly dominated by (df= 1;°), and (duodd;°)is weakly dominated by (du+ 1;°).
T hroughoutthe paperthe proofs related to weak dominance results are similarto the oneofL emma2 and,henceforth,theseproofs areomitted.W hatwillbetheequilibrium outcomes?
L emma4 IfW 0 <
1+±¡°
1+± then the strategies(du¸4 and even;°)and(du=1;¢)are weakly dominated.
L emma5 IfW 0 >
°
1+± then thestrategies(df¸3andodd;°)and(df=1;¢)areweakly dominated.
6
T he sam e resultsc anb e ob tained using rationaliz ab ility c oncepts(d e¯ned inHeringsand Vannetel-b osch [6],[5], and VannetelVannetel-b osch [12 ]) instead ofequiliVannetel-b rium onesfor b oth the b argaining gam e and the d ead line gam e.
7T he d iscount fac tor c analso b e expressed b y the formula ±= exp(
¡r¢¢ ), w herer > 0 isd iscount rate and ¢ isthe length ofa single b argainingperiod .
Could itbe thatthe no deadline situation is an equilibrium outcome? T he answer is negative. Indeed, from L emma 2 to L emma 5 we already know that, the ¯rm will choose (1;°)forsure ifW 0 > 1+°± and the union willchoose between (0;°)and (2;°)if W 0 <
1+±¡°
1+± .InordertofullycharacterizetheT H P E outcomewedistinguishthreecases.
Case1 :1+1+±¡±°> 1+°± ¸W 0.
Itfollows from L emma2 toL emma5 that, atequilibrium,the union could choose eitherthe strategies (0;°)or(2;°)and the ¯rm could choose eitherthe strategies (df¸1 odd;°)or(1;¢).Iftheunion chooses (2;°)then theuniquebestresponse
forthe ¯rm is (1;°). Ifthe union chooses (0;°)then bestresponses forthe ¯rm are (df ¸1 odd;°)and (1;¢).Ifthe ¯rm chooses (df ¸3 odd;°)or(1;¢)then
the unique bestresponse forthe union is (2;°).Ifthe ¯rm chooses (1;°)then the unique bestresponse forthe union is (0;°)ifW 0 < (1+±)(1± ¡±)(° ¡°)and is (2;°)
otherwise. Indeed,W ¤(0;°)= ±
1+±° > (1 ¡±)W 0 + 1+±±° =W ¤(1;°)reverts to
W 0 < (1+±)(1± ¡±)(° ¡°). A s a consequence, the unique T H P E isf(1;°);(0;°)gif W 0 < (1+±)(1¡± ±)(° ¡°)and isf(1;°);(2;°)gotherwise.
Case2 :1+1+±¡±°> W 0 >
°
1+±.
Itfollows from L emma2 toL emma5 that, atequilibrium,the union could choose either the strategies (0;°)or (2;°)and the ¯rm is going to choose for sure the strategy(1;°).H ence,giventhatthe¯rm chooses (1;°),theunionwillchoose(0;°) ifW ¤(0;°)=1+±±° > (1¡±)W 0+ 1+±±° =W ¤(1;°).T hatis,ifW 0 < (1+±)(1¡± ±)(°¡°).
O therwise,theunion willchoose(2;°).
Case3 :W 0 ¸1+1+±¡±°> 1+°±.
Itfollows from L emma2 toL emma5 that, atequilibrium,the union could choose either the strategies (0;°)or(du¸2 even;°)or(1;¢)and the ¯rm is going to
choose forsure the strategy (1;°). H ence, given thatthe ¯rm chooses (1;°), the union willchoose (0;°)ifW ¤(0;°)= 1+±±° > (1 ¡±)W 0 + 1+±±° =W ¤(1;°).T hat
is, ifW 0 < (1+±)(1± ¡±)(° ¡°).O therwise, the union willchoose (du¸2 even;°)or
(1;¢).
H avingcharacterized the equilibrium strategies we can showthatthe deadline stage game has a unique T H P E.Indeed, ifW 0 > (1+±)(1± ¡±)(° ¡°)then the unique T H P E
outcome isd= 1 and° =°.So, atequilibrium, the ¯rm and the union willstartthe wage bargainingunderthe threatofa severe lockoutatperiod 1 .T he ¯rm willmake a wageo®erW ¤(d= 1;°)= (1 ¡±)¢W 0 + 1+±°± atperiod 0 ,and theunion willacceptthis
P roposition1 IfW 0 > (1+±)(1± ¡±)(° ¡°)then the negotiation modelhas a unique and e±cient equilibrium outcome. T he deadline (a lockout threatof a strong intensity) is settledatperiod1 andan agreementonW ¤(d= 1;°)is reached immediately atperiod0.
H owever,ifW 0 < (1+±)(1± ¡±)(° ¡°)thereis auniqueT H P E strategypro¯lewherethe
¯rm chooses (df= 1;°)and theunion chooses (du= 0;°).So,ifW 0 < (1+±)(1± ¡±)(° ¡°)
the unique T H P E outcome isd= 0 and° =°, and itis an ine±cientoutcome since a strikeofaweakintensityoccurs immediatelyatthestartofthewagebargaining.
T hatis, ifthe equilibrium wage in case ofan immediate strike ofaweak intensity is greaterthan theequilibrium wagein caseoftheunion movingatthedeadlineand facing thethreatofalockoutofastrongintensity,then theunion willchooseatequilibrium to implementimmediately a strike ofa weak intensity. W hich is the intuition behind this result? Since the old wage contractis smallenough, the di®erence between° and° is largeenough and theplayers are enough patient,itbecomes optimalfortheunion togo immediatelyon strikeandtodestroypartoftheavailablesurplus.Indeed,such acon° ict ofaweakintensityallows theuniontoavoidhavingtoacceptaverylowwageo®erfacing the threatofa severe lockout, where the ¯rm would grab mostofthe surplus. So, at equilibrium,theunion goes intocon° ictimmediatelyand aP areto-dominated agreement follows.Infact,this equilibrium outcomeis P areto-dominatedbytheequilibrium outcome ofB (du= 1;°).A tequilibrium, the ¯rm willmake awage o®erW ¤(d= 0;°)=1+±°± at
period 0 ,and theunion willacceptthis o®erimmediately.
P roposition2 IfW 0 < (1+±)(1± ¡±)(° ¡°)then the negotiation modelhas a unique and ine±cientequilibrium outcome.A con° ictofa weakintensitystarts immediatelyattime
0 followed by an immediate agreementonW ¤(d= 0;°).T he per-period e±ciency loss is equalto1 ¡°.
So,the¯rm atequilibrium chooses (df= 1;°)becauseeitheritis adominantstrategy
orifhedoes notthen theunion would chooseanotherdeadline and itwould bethe¯rm that would face the threat ofa severe strike where the union would grab most ofthe surplus.
T heconditionW 0 < (1+±)(1± ¡±)(°¡°)is satis¯edwhenevertheoldwageW 0 isrelatively
small,eachplayerhasathisdisposalbothactionsthatreducesubstantiallythevalueadded in the future and actions thathave only a minorimpacton the future value added8 (in
8
T he ac tionsasw ellastheir im pac t onthe future value ad d ed m ay d epend onfac torssuc h asthe c om petitiononb oth the prod uc t m arket and the lab our m arket (see Cutc her et al.[1]).Ind eed , inc ase there isa strong c om petitiononthe prod uc t m arket, a lab our c on°ic t m ay ind uc e a b ig lossinm arket pow er and future revenuesto b e generated .How ever,inc ase there isa strongc om petitionand m ob ility on
other words, the di®erence between° and° is large enough)and the players are not impatient(±is largeenough).
B eforeconcludingwewillalsoconsiderthelimitcaseoffullypatientplayers as±goes toone.
Corollary1 A s± goes to one or as the intervalbetween o®ers and countero®ers van-ishes,the negotiation modelhas a unique andine±cientequilibrium.A con° ictofa weak intensitystarts attime zero followed byan immediate agreementonW ¤(d= 0;°)=12°.
N otice that we have taken the limit± ! 1 assuming that the players can stillre-duce permanently the future value added.9 T his assumption may be questionable once we reinterpretthe limitas the intervalbetween o®ers and countero®ers is vanishing.A n alternativeassumptionis tosupposethatthelevelofsurplus destructionwouldbedeclin-ingwith ¢ decreasing.N evertheless, we stillobtain the ine±ciency resultifwe assume thattheintensities ofacon° ictarebounded belowone.T hatis,if°(¢ )< °(¢ )< ±for all¢ and lim¢!0 °(¢ )< lim¢!0 °(¢ )< 1 .
4
Con
c l
usion
W ehavedeveloped atwo-person negotiation modelwith completeinformation which is a ¯rstattempttomake endogenous both the deadline and the levelofsurplus destruction afterthe deadline. W e have shown thatthe equilibrium outcome is always unique but mightbeine±cient.M oreover,as the bargainingperiod becomes shortoras the players become very patient, the unique outcome is always ine±cient. So, ourmodelmay also justifytheexistenceofP areto-inferiorphenomenaotherthanlaborcon° icts,suchas tari® wars, debt moratoria, break-up ofcease-¯res or wars in general. O ne very interesting extensionwouldbetoconsideramoregeneralbargainingprocedureas inP erryandR eny [8],which allows theplayers tochoosewhen and whetherornottomakean o®er.
R ef
eren
c es
[1 ]Cutcher-G ershenfeld,J.,Kochan,T .andJ.CalhounW ells,1 998,"H owD oL aborand M anagementV iewCollectiveB argaining,"M onthlyL aborR eview1 21 (O ct.),23-31 .
the lab our m arket,the im pac t ofa c on°ic t c ould b e red uc ed since the ¯rm m ay¯nd m ore easilytem porarily replac em ent w orkers.
9Inc ase the c on°ic tshave anad d itive d estruc tive im pac t onthe future value ad d ed ,a su± c ient c ond ition suc h that, as±! 1 the T HP E outc om e isunique and ine± c ient, isthat the intensity ofthe strongest c on°ic t islarge,i.e.°issm allenough.
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