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American Economic Association

Wage Bargaining and EmploymentAuthor(s): Ian M. McDonald and Robert M. SolowSource: The American Economic Review, Vol. 71, No. 5 (Dec., 1981), pp. 896-908Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1803472 .

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WageBargainingand EmploymentBy IAN M. MCDONALDAND ROBERTM. SOLOW*

One of the perennialproblemsof businesscycle theory has been the search for a con-vincingempiricaldescriptionand theoreticalexplanationof the behavior of wage ratesduring fluctuations n output and employ-ment. Even the empiricalquestionis hardlysettled, although the most recent carefulstudy (see P. T. Geary and John Kennan)confirmsthe prevailingview that real-wagemovementsare more or less independentofthe businesscycle.Thereare reallytwo sub-questionshere.Thefirstpresumes hat nomi-nal wage stickiness is the main route bywhichnominaldisturbances ave realmacro-economic effects, and asks why nominalwages shouldbe sticky.The second focuseson realwages,and asks why fluctuations nthe demand or laborshould so often lead tolargechanges n employmentand small,un-systematic, hanges n the realwage.We addressonly the secondof these sub-questions.Wedo so in the contextof explicitbargaining verwages and employmentby atrade union and a firm or group of firms,thoughone couldhope that the resultsmightapply loosely even where an informallyorganizedaborpool bargainsmplicitlywithone or more ong-timeemployers.We do notharbor he illusionthat tradeunionsare theonly important source of wage stickiness.There are other plausible(and implausible)stories.Some, like this one, rest partiallyonoptimizingdecisions;othersdo not.The impulse to this study was macroeco-nomic,but ourfocusis on a singleemployerand a single labor pool. Our methods, andthereforeourconclusions,areentirelypartialequilibrium.f theshort-runmobilityof laboris slight,and if fluctuationsn realaggregatedemandaffect many sectorssynchronously,thenperhaps hemechanismwe uncoverherecould be important in the business cycle

context. But the work of embedding t in acomplete macroeconomicmodel remains tobe done.We begin with a model in whichthe unionis a simplemonopolist,settingthe wagerateunilaterally o maximize heexpectedor totalutilityof its members,and allowing the em-ployercompletediscretionoveremployment.We then consider a more complex institu-tional setupin whichthe unionand the firmare supposedto bargainoverboth wage andemploymentand reach an outcomeefficientfor them both. (The monopoly outcome isnot efficient, for the traditional reason.)There s, of course,a wholerangeof efficientbargains.A completetheorymust single outone of them,but there is unlikelyever to beuniversalagreementon the rightway to doso. Ourapproach s simplyto try out severalsimple conventionsand several formalsolu-tions to the bargainingproblem.We providea frameworkwithin which they are all seento bear a familyresemblance o one another.Moreover, there is a certain assumptionwhichmakesall theproposedsolutionssharean importantcharacteristic:he effects of adownswingor upswingin final demand onthe negotiatedoutcome can be decomposedinto two steps which reinforceeach otherwith respect to employmentand offset eachother with respectto the wage. So it wouldnot be surprisingo findlargefluctuations nemploymentand smallunsystematic luctua-tions in realwagesduringbusinesscycles.Thekeyassumptions thatproduct-marketconditionsare moresensitive o the businesscyclethanthereservationwage s. This wouldbe the case, for instance, if (a) nonmarketopportunities including unemploymentin-surance benefits, which are not cyclicallyvulnerable, play an important role in thedetermination f thereservationwage,and/or(b) interemployermobility s so limited thatoutside market opportunities figure onlyslightly n workers' alculations.*University of Melbourne and Massachusetts n-stituteof Technology, espectively.

896

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to different degrees by aggregate fluctua-tions. For thisreasonalone-as a refereehaspointed out-one mightexpect the relevantreservationwage, andeven its averageacrossindustries, to vary systematically duringbusiness cycles. Nevertheless, o the extentthat industrialand occupationalmobilityislimited in the timespan relevant o businesscycles, we believe our story retains plausi-bility.The bestdevelopedanalyticalapproach othis problemis the theory of implicit con-tracts surveyedby CostasAzariadis). n thatliterature,a contract is a long-termagree-mentin the sense that the economicenviron-ment will change n an only probabilisticallyknown way duringthe life of the contract.On the reasonableassumption hat the firmis less risk averse than its employees, thetypicaloutcome s that an efficientcontractwill be wage stabilizingand (unless specialfeaturesareintroduced) mployment tabiliz-ing as well. In ourapproach,by contrast, hewage-employmentbargain is struck aftereconomic conditions in the firm's productmarketand in the surroundingabormarketare known. It is a short-termor one-shotcontract.Risk entersonly in a trivialsense:if the contractcalls for a fraction of theunion'shomogeneousmembershipo be un-employed, the unlucky ones are chosen atrandom.In real life, negotiatedcontractsare usu-ally long term. But they do not specifyem-ployment,which typicallyis left to the dis-cretion of the employer; in consequence,employment luctuatesa lot. Our reconcilia-tion of the stylizedfacts, the theoryof im-plicitcontracts,and our owntheorygoes likethis: if a series of shortcontractswouldleadas our model suggeststo wide variation nemployment ndfairlystablewages,thenthesame outcome might reasonably well beachievedby a long-termcontract n which astable wage is specified but the level ofemployment s chosen at will by the firm.(We do need a generalrestraint n employersof a sort that could be accomplishedby"featherbedding"orkrules.)

I. ASimpleMonopoly nionThe simplest interesting noncompetitiveinstitutional setup is that of a monopolyunion which can set the wage unilaterally.The employer(or employers)then choosesthe volumeof employment.Most collectivebargainingagreementsdo give the employerdiscretionover the volume of employment.Whythis shouldbe so is an interestingques-tion (see RobertHall and David Lilien). Butit is a raretrade union that literallycontrolsthe wage and we take up more complicatedbargainingarrangementsater. The simplemonopoly case has been analyzedbefore,of

course(see, for example,Allan Cartter),andwe haveonly a few noveltiesto add. We usethis analysismainly as a vehicleto introduceconcepts, establishnotation,and drawsomediagrams.The firm is characterizedby a revenuefunction R(L) giving sales proceeds as afunction of employment. f the firm were apricetaker n itsproductmarket,R(L) wouldbe simplypF(L) wherep is the parametricproduct price and F(L) is the productionfunctionrelatingemployment o output.Weassume,as usual, that R(O)=Oand R(L) isconcave, with marginalrevenue eventuallybecomingverysmallor evennegative.Profitis then R(L)-wL.' If the firm is a profitmaximizer,it is indifferent among (w, L)combinations that leave R(L) - wL constant.These isoprofit curves in the (w, L) planeserveas indifference urves or the firm.Theslope of an isoprofitcurvethrough w, L) isdw/dL=(R'(L)-w)/L. For any L, iso-profit curves have positive slope until wreachesR'(L), then negative. For higherL,the switchoccursat a lowerw, so the firm'sindifferencemap is as shown in Figure 1.For any L, a smaller w creates a biggerprofit,so lower soprofitcurvesarebetterforthe firm.Let the union quote a wage w1. The firmthen seeksthe lowest indifference urve thattouchesthe horizontal ine at heightwl. Thatis to say, it solves R'(L1)-w, =0: marginal

'Product price andwage rateare to be thoughtof asdeflatedby a generalprice ndex.

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LD

LFIGURE 1

revenueproductof labor equals the wage. Inother words, the firm's demand curve forlabor is the locus of maximumpoints of theindifference urves n the (w, L) plane. It isdownward loping,by virtueof theconcavityof R(L).The union can achieveany pointalongthefirm's demand curve. What is the union'sobjective?That is an old questionin laboreconomics. We choose a particularansweranduse it throughout. uppose heunion hasN members, all alike. If L of them areemployed,eachmemberhasprobabilityL/Nof having a job and achievinga level ofutility U(w)-D and probability1-(L/N)of not beingemployedby the firm,whereDis the fixed additivedisutilityof holding ajob.2 If not employedby the firm a workerachieves a level of utility U(%w), where Wcan be thoughtof, forshort,as an unemploy-mentcompensationbenefit,but shouldreallyinclude all the other contributions to thestandardof livingthat wouldnot be receivedif the workerwereemployedby the bargain-ing firm. U(x) is the standardsort of con-caveutilityfunction.

LD

A

LFIGuPuE2

The expected utility of a union memberis therefore N -'{L(U(w)-D) + (N-L)U(w,)}, which can be writtenas U(w,)+N-'L{U(w)-D-U(w%)}. Since WU nd Nare treatedas data for the purposeof unionwage setting, we can set D+ U(w,)= U andsummarizeby saying that the union wishesto maximizeL(U(w)- U). The logic of thisis that L(U- U) is the membership's ggre-gate gain fromemployment,over and abovethe incomew%hat everymember tartswith.The union's ndifferencemapis derived romL(U(w) - U) = constant; the indifferencecurves have the usual downward-slopingconvex shape in the (L, w) plane. They havethe special property hat they are all asymp-totic to the horizontalat w= w-,wherew isderived from U(w) =U. This is shown inFigure2.

The best wage for the union to set isdetermined in the obvious way by thetangencyof an indifferencecurve with theemployer's abor-demand urveas shown inFigure 2. Mathematically, his amounts tofinding the maximumof L(U(w)-U) withrespectto L and w, subjectto the constraintR'(L)-w=O. We can write down the first-orderconditiondirectlyby equating heslopeof the indifference urvethrough w, L) (i.e.,-(U- U)/LU') to the slope of the (inverse)demand function (i.e., R"(L)). Since w=R'(L) at any eligible point, the first-order2We gnore-by choice-the possibility hatworkersarefree to choosethehoursand intensityof work.

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conditioncan be writtenas(1) -LR"(L)/R'(L)

= (U(w)-U )/wU'(w).The left-hand side is the reciprocalof thewage elasticityof thedemand or labor,takenpositively;the right-hand ide is the recipro-cal of the elasticityof the gain from employ-ment (U- U) with respectto the wage. Sothe condition is that the two elasticitiesshould be equal. (There is a second-orderconditionthatwe assumeto be satisfied.)What is the natureof wage behavior m-plied by this model?A change in demandconditionswill affect the union'swage deci-sion via two routes,the elasticityof demandfor laborand iw.We consider hem in order.Solve w=R'(L) to give the demandfunc-tion in directform,andinsert a parameterB(for businesscycle),with the convention hatan increasein B increases the demand forlabor at any wage. Thus the demand forlabor is L=G(w, B). As B rises and falls,how is the effectdividedbetweenchanges nw and changes n L? Consider he first-ordercondition(1) writtenas(2) wGw(w,B)/G(w, B)

-WU'(w)/(U(w)-U).The cyclical sensitivityof the wage clearlydependson the extentto whichchanges n Baffect the elasticityof demandfor labor atany givenwage.For instance, f the demandfunctionshifts isoelastically- that is, the de-mandfor laborfalls in a recession,but withits elasticityunchangedat each wage-thenwe can alwayswrite G(w, B)=BG(w), anditis obviousthat (2) does not dependon B atall. In thatcase,thewagewillbe rigidduringbusinesscyclesand fluctuationswill fall en-tirelyon employment.One can easily imag-ine cases in which the monopoly wage willmove countercycically,or procyclically orthat matter,thus diminishingor magnifyingthe accompanyingfluctuationsin employ-ment.

The other way in which the level of ag-gregateactivitycan affect the desiredwageis

through wT,which is composed of severalelements. Some of these elements, such asunemployment benefits, the value of leisure,the value of working around the house, netgains from illegal activities, would appear tobe affected very little, if at all, by aggregateconditions. (Unemployment benefits aresometimes raised in recession, imparting anupward effect on w- and thus w.) But theother major element in w is the expectedvalue of alternative employment opportuni-ties and this should have a strong procyclicalfluctuation through changes in the probabil-ity of finding alternative jobs and in theirwages. The effect this has on the wage ratewill depend on just how important a compo-nent of w- t is. If job mobility is low, and/orif changes in wage rates in other jobs aresmall, then the effect of changes in alterna-tive job opportunities will be slight.We conclude this section with a canonicalexample. Let f be a constant elasticity ofdemand for labor, however generated. TakeU(w)=wb/b, where b is less than one, butmay be negative. Then (1) yields w/lw=-(I -b/f ) -l/b. Thus the monopoly wage dependsnegatively on the elasticity of labor demandand negatively on the risk-aversion parame-ter (1 -b). Intuitively this is how it shouldbe. For example, if f= 2 and (1 -b) = 3, thenthe monopoly wage is (5)1/2 times the"minimum supply price" wi. If f is as low as4, W= 3w. If f= 2 and (1-b)= 2, w= 3w. If

f= 4, (1-b) = 2 then w= 5w. These low val-ues of f are in accord with econometric re-sults. Notice that if they are combined withpositive values of (I - b) less than 1 theoutcome is much less "realistic": thus f= 2and 1-b 2/3 implies w= 27W.II. EfficientBargains

The model of wage determination just de-scribed is even more like simple product-market monopoly than it looks. The dif-ference in appearance arises because the mo-nopolist, in this case the union, maximizes autility function and not profits. It is notsurprising, then, that the wage-employmentoutcome shown at point A in Figure 2 is notefficient. There are wage-employment pointsat which both parties are better off. This is

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900 THE AMERICAN ECONOMIC REVIEW DECEMBER 1981

D Contract IL ~~~~~~~Cuve

wi~~AW

W K~~~~-L N

FIGURE 3

easily seen in Figure3. The constant-profitcurve passing throughA is by constructionhorizontalat A. Therefore t cuts the down-ward-sloping ndifferencecurve throughA.Theregion o the southeastof A, between heisoprofitcurve and the indifference urve, sthe set of outcomesPareto uperior o A. Themonopolywage is too high and employmenttoo low. Obviously efficient bargains arepointsof tangencybetweenan isoprofitcurveand an indifference urve.We call the locusof such points the contract curve; in thiscontext that is the mot juste. An example sshown n Figure3.More complicated institutional arrange-ments are necessaryfor the achievementofefficient bargains.In particular, he unionhas to exercise some sort of influence overthe level of employment, n contrast to thesimplecasewherethe level of employmentsset unilaterallyby the employer. Since theobjective s to increaseemploymentbeyondthe level given by the labor demand sched-ule,manningagreements r "featherbedding"are likely to be adopted.If it is impracticalto specify the level of employmentin thecontract,an efficient outcome may be ap-proximatelyachievable f the contractspeci-fies the numberof workersper machine,orsome other similarrule,and leavesthe over-all aggregateto the discretionof the em-ployer.

The contract curve is characterizedbyequalityof the slopes of a unionindifferencecurveand an isoprofit curve.This conditionyields the equation(3) (U(w)-U(w-))/U'(w)=w-R'(L).The first thing to notice is that the contractcurve intersects the firm's labor demandcurveat w= iw,because heright-hand ide of(3) vanishesalongthe demandcurve,and theleft-handside3 is zero only at iw.The point(iw-,L) is actually the competitive outcomefor thismodel.If therewere no union and Uwere the level of utilityattainableelsewherein the economy,4 hen iwwould be the givensupply price of labor to the employer,whowouldmaximizeprofits at L.The slope of the contract curve is, bydifferentiation f (3),

dw/dL - U'(w)R"(L)/(U"(w)(R'(L) -w)).

Thus thecontractcurve s momentarily erti-cal at (iw,L), andpositivelyslopedelsewhere(because, rom (3) w' i impliesw'R'(L)).sNo bargaincan be struckwith w

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thus being induced,presumably y an all-or-none offer, to employmore workers han itwould like at the agreed-uponwage. This isthe insight that led to Wassily Leontief'spioneering paper. An even stronger state-ment is true: all along the contract curve,exceptat (iw,L), the marginal evenueprod-uct of employment alls short of W.If onethinks of iwas the true supply price of laborto the employeror industry,then this is astrongreminder hat the bargainsalong thecontract curve are efficient only from thepoint of view of the employerand the fixedmembership f the union.To see how the contractcurve is affectedby changes n the economicenvironment,werewrite he revenue unctionas R(L, B), andassume that RB and RLB are both positive:prosperity increases total revenue and themarginal revenue product of labor at anylevel of employment.Then (3) becomes(3') (U(w)-U(wi))/U'(w)

=w-RL(L, B).If we now differentiate 3')withrespect o

B, holdingL constant,we findaw/aB=RLB(L, B)U'(w)2

/((U(w) - U(w)) U"(w))

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the union acting as a family or commune,pooling all earningsand redistributingn-come from its employed to its unemployedmembers,so that they all have equalutility.Specifically, uppose he union pays outYe oeach of its employedmembersandyuto eachof its unemployedmembers.Thesepaymentsare connectedby(4) U(yu)= U(ye)-D;the employedworker s compensated or thedisutilityof work. (For any old-timerswhoremember he ArtYoung cartoon:"Meslav-ing overthis hot stove and you working n anice cool sewer," t is perfectly all right tothink of D as negative.)The union can onlypay out what its memberspay in, so there sa second constraint(5) Lye+(N-L)yu=Lw+(N-L)wu;the employedcontribute henegotiatedwageand the otherstheirunemployment enefits.Sinceeveryone s equallywell off expost, theaggregateutility functionis simplyNU(yu).Here(4) and (5) can be solvedforYeandyuas functionsof w and L, so that the collec-tive utility function can be thoughtof as afunction of the negotiatedoutcome as be-fore.Straightforwardifferentiationof (4) and(5) leads to the slope of the union's indif-ferencecurvethrough w, L):

dw/dL -L-l[(w-wu)-(YeeYu)].Thisexpressions negativebecause heunionactuallyredistributes ash income from theemployed o the unemployed.The locusof tangenciesbetweenthe indif-ference curves and the isoprofit curves de-fines a new contractcurve.Its equation s

R'(L) =ye Yu +wu,whereYeandyuare functionsof w andL. Weomit the detailsbut record hat this contractcurve is downward sloping, passes through(w, L), and lies to the rightof the demandcurve for labor. The picture is thus as inFigure 5.

wLD ContractCurve

LFIGURE 5

It remainstrue that the wage exceeds themarginalrevenueproduct of employment nefficientbargains.The employerwould pre-fer to reduce employmentat the negotiatedwageif thatwerepermitted.But employmentat a given wage is smallerin the setup ofFigure5 than it is at the same wage in thesetup of Figure3. The reason is that the expost equalizationof Figure5 diminishes heincentive of the individual member to beamong the employed.One importantconse-quenceis that in Figure5 the marginalreve-nue product of employment exceeds thesupply price of labor at every efficientbargain.

IV. SomeSimpleConventionsMost formaltheoriesof bargaining ssumethat the negotiatedoutcome will lie on thecontractcurve, except perhapsfor the occa-sional conflict-a strike,say-when bargain-ingbreaksdown.Wehave some doubtsaboutthe empiricalrelevance of this assumption.But it is hard to see how one could proceedwithout it; so we will use it tentatively,withan eye out forits compatibilitywith commonobservation.Even so, as we have said, thereis no generallyaccepted rule for selectingapoint or other small subset of the contractcurveas an especially ikelycandidate or the

actual negotiated outcome. (The book byGeorgede Menil containsan excellent, airly

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recent,summary f the state of the theoryofwage bargaining.Models of auctionmarketsor sealedbid procedures ardly eemto applyin this context.)In this section,we considerafew very simple conventions, any one ofwhich might seem plausible in a specificcontext,but none of which has any seriousclaim to generality.In the next section wewill take up a couple of formalbargainingmodels which do make such a claim. Afterthat, we turnto conventions hat mightapplyto the renegotiationof an original bargainwhen the environment hanges.Throughout,we emphasizethe "businesscycle" implica-tions of each solution, not its place in thetheoryof bargaining.

A. A Dominant Union and "Fair Shares"Points to the northeastalong the contractcurve are successively ess profitable or theemployerand morefavorable o theworkers.A powerfulunion might be able to force thefirm to accept zero profits,if we take zerosomewhat arbitrarilyas the level of profitbelow which the firm would leave the in-dustryor shut down. Thatsuggestsadjoining

to the equationof the contractcurve(3') thezero-profit onditionR(L, B) wL. Geomet-rically speaking, this hypothesissingles outthe point at which the contractcurveinter-sects the zero-isoprofit urve.The hypothesis of a zero level of profitscan easily be generalizedand made less ex-treme.Suppose that historyhas led to thenotion that there is a "fair"division of netrevenue between the workers and the em-ployer.If the normalshareof wages is 100kpercent,we can write(6) wL=k R(L, B).The caseof zeroprofits s simplyk 1. Now(3') and (6) are the two equations definingthe negotiated wage and employment. Ex-cept whenk= 1, (6) does not coincidewith aparticularsoprofit ocus.)The contract curve (3), as we know, isupwardslopingin the (w, L) plane; (6), onthe otherhand, representsw as a fractionkof the averagerevenueproductof laborandslopes downwardby our assumptionson R.

This patternwill repeat tself: the negotiatedoutcome is at the intersectionof an upward-sloping efficiency locus and a downward-sloping locus that can be interpretedas re-flectingequity(or power)considerations.Suppose the economic environmentde-teriorates in a recession. If the product-marketeffect dominates he labor-marketf-fect,we can concentrate n a reduction n B.The contractcurveshifts to the left, as shownearlier.The locus (6) shifts down. From thecrude geometry t is clear that employmentmust fall, but the negotiated wage can goeitherway.Thatis a promisingbeginning ora wage-stickiness tory, so we work out theresult exactly.Differentiationof (3') and (6) leads in theconventionalway to:

( z -R LL \ (dw/dB (RLB 'VL w-kRLj dL/dBj 1kRB/where z stands for d/dw[w - (U(w) -U(w))/ U'(w)] = ((U(w) -U(w)U"w))/U'(w)2.The determinanthas sign pattern(114+)nd is thereforenegative.Calculation,and substitution of the value of k from (6)shows that(7) sgn dw/dB -sgn {RLB(1-LRL/R)

+LRLLRB/R}Thisdoesnot depend explicitlyon the utilityfunction, except as it helps determine thepoint at whichR is evaluated.The first termis positiveand the secondnegative,confirm-ing the indeterminacy f the sign of dw/dB.Two specialcases are worthnoting. Firstof all, supposeR(L, B) canbe written n theform BS(L/B); this gives rise to the iso-elastically shifting labor-demand urve dis-cussed earlierin connectionwith (2). Thendw/dB=O, always.So whenever he elastic-ity of demandfor labor at the going wageisapproximatelynvariant o the businesscycle,the wagewill be sticky.The second special case puts R(L, B)=BS(L); this makesthe inversedemandcurvefor labor shift isoelastically n the businesscycle so that the demand elasticity at the

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going levelof employments invariant.Thendw/dB is opposite in sign to d/dL(LRL/R),which again suggeststhe lack of any strongdirectionalityn the businesscycle.If it were the case that firms typicallybecome sales constrained n recessions, sothat revenueelasticity alls, the model wouldindicate a countercyclical ise in the wage.That seems extreme;but perhapsone mightconcludethat efficientbargainingwill makeemployment,more than the wage, bear thebruntof cyclical adjustment.We do not exploredL/dB in detail, be-cause it is obvious from the geometrythatthis model makes employment stronglycyclical.

B. A Dominant EmployerIf the unioncalls the tune,it is limitedinits demandsby the possibility that the firmwill shut down. If the employer calls thetune, thereis (usually)some similar imit tocomplete freedom of action. It may comefrom the possibilityof a strikeor other dis-ruption,or it may come from the needof theemployer o preservea laborpool whenthere

are opportunities or employmentelsewherein the economy.Even a dominantemployerwill push only so far to the SW along thecontractcurve.We can imaginethat thereisan indifferencecurve below which the firmwill not wish to push its labor pool. There sa conceptual hoice here:we couldtakesuchan indifference urve to be givenby L(U(w)- U(wi))=constant, that is, by the gain tothe workers rom membershipn the firm'slaborpool. The significanceof this choice isthat if w- alls, the firm can lower its wage atgiven employment o keep the workers'gainconstant.On the otherhandif we had fixedthe limitingindifferencecurve by LU(w)+(N-L)U(w-) = NU(W-)+L(U(w) -U(w-))constant, he firmwould have to increaseitswage offer to make up for a reduction n w.That seems rather too paternalistic or reallife.In this excessively paternalisticcase, infact, a simultaneousreductionin iwand Bmustalways ead to a highernegotiatedwagealong the new contract curve. Under thealternativeassumption,as usual, there are

forces working n both directions.The reduc-tion in B pushes the negotiated wage up-ward; the contract curve shifts to the NWand so does its intersectionwith the un-changed imiting ndifference urve.A reduc-tion in wi lowers the limiting indifferencecurve at given employment more than onefor one, in fact) and thus pushesthe outcometo the SW. Generalizedrecession thus re-duces the employment side of the bargainunambiguously,but the wage can go eitherway.V. FormalBargaining heory

Most formal theories of the bargainingprocess proceed axiomatically.Usually oneof the axioms is that the bargainedoutcomeis efficient. But then, instead of arguing hatthis or that outcomeon the contractcurve smore "natural" han others, the bargainingtheoristproposes desirable propertiesfor arule that would permit a referee equippedwith it to go from one bargainingproblem oanother n some broadclass, and produce asolution to each one by applicationof therule. (One of the desirable properties, ofcourse, s that the rule shouldalwayschoosea point on the contractcurve.)The goal ofthe theoristis to find a set of plausibleoracceptablepropertiesand show that there isonly one rule with those properties.Raiffaargued early on that such solutions of thebargainingproblemmightbestbe thoughtofas ArbitrationRules; they might not havemuch descriptivevalidity in predictingtheoutcomeof rawbargaining,but they providea defensible handbook for an arbitratorwhosejob is precisely to settle a streamofbargaining onflicts.Fromourpointof view,Raiffa's interpretation s perfectly accept-able.Formal theories usually operate not interms of the contract curve but in the"bargainingset" and its efficient frontier.The bargaining et is relatedto the contractcurve in exactly the way that a productionpossibilityset is relatedto the contractcurvein a productionbox or a utilitypossibility etis related to the contract curvein an Edge-worth exchangebox. To beginwith, we needto construct hebargaining et for ourmodel

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R(L)- w L (V*G*)

I F IE

NU W LU(;W)+(N-C)UiW)FIGURE 6

of wage bargaining.Each possible outcome(w, L) corresponds o a payoff to each party.In the case of the employer, the payoff isG=R(L)-wL; to the union, the payoff isV=LU(w) + (N-L)U(W). The set of (G, V)swept out as (w, L) rangesover its possibili-ties is the bargainingset. If no bargainisstruck,we take t thatG=06 and V=NU(w).As (w, L) traverseshe contractcurve, G,V)traverseshe undominated fficientsubsetofthebargaining et, by construction.Axiomaticbargainingheoriesrequire hatthe bargaining et be convexso that, in theusual way, the efficient payoff-possibilitycurve s a decreasing oncavefunction n the(G, V) plane. They justify an assumptionofconvexity by the possibility of randomiza-tion. That would hardly do in the wage-bargainingcontext, but fortunatelythe as-sumptionswe have made on R(.) and U(.)guarantee,as tedious calculationwill show,that the frontierof the bargaining et is wellbehaved.The picture,therefore,s as in Fig-ure 6. (Here wi7s the highest wage theemployercan pay and still break even at apoint on the contractcurve;see Figure 3.)Selectionof a point on the contractcurveisequivalent to selection of a point on theefficiency ocus.The best-known formal solution to thebargainingproblem s Nash's. It selects the

efficient point that maximizes heproduct ofthe parties' gains over and above the no-contractoutcome.In this case, it maximizesG -(V-NU(w)) or (R(L)-wL)(U(w)-U(w))L overthe bargaining et. In principle,one mightthink of maximizing hat productsubject o the equationof the contractcurve;but on reflection, the constraint can beomitted.Unconstrainedmaximization f theproductby choice of (w, L) will certainly ryto maximizeR(L)-wL for any given valueof L(U(w)- U(W)), so the equation of thecontract curve will reappearas one of thefirst-orderconditions of the unconstrainedproblem.It does. The other first-order ondi-tion turnsout to imply,ratheroddly,(8) w= (R/L+R'(L))/2.At the Nash solution, the wage is equal tothe arithmetic mean of the average andmarginal evenueproductsof labor!The Nash solution is thus definedby (3)and (8). Under our assumptionsabout R(.),both the averageand marginal evenueprod-ucts are decreasing.So (8) defines a nega-tively sloped "equity" ocus that intersectsthe contractcurveonce,at the Nash solutionto the bargainingproblem. Once again, wecan replaceR by R(L, B) and ask howvaria-tion in B affects the wage coordinateof thesolution. Upon calculation, t turns out thatthe criterion 7) holds heretoo, andso do theparagraphs f textimmediatelyollowing 7).One of the axioms leading to the Nashsolutionof the bargainingproblemrequiresthe rule to be "independentof irrelevantalternatives." upposethat E is the solutionto the bargainingproblempictured n Figure6; now define a new bargainingproblembydeleting part of the bargaining et, any partso long as the point E remains.The axiomrequiresthat E be the solution of the newbargainingproblem.Since the deleted out-comeswerenot chosenby the rule whentheywereavailable, hey are"irrelevant"nd theirabsence should make no differenceto theoutcome. This axiom has been much com-plained about, and justly. Intuitions about"bargaining ower"and "fairness"mightin-cludethe notion thatif A couldwin a lot in abargaining ituation,he or she is "entitled"6It would not be hard to allow for fixed costs F, sothat G= - F if no agreements reached.

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906 THE AMERICAN ECONOMIC REVIEW DECEMBER 1981to morethan if he or she could only, in thebest of circumstances,win a little. Anyonewho shares that intuition does not believethat "irrelevant"lternatives reirrelevant.Dissatisfactionwith this axiomhas led tootherdefinitionsof the solutionto a bargain-ing problem.Ehud Kalai and Meir Smoro-dinsky proposereplacingthe unsatisfactoryaxiomwith another.Startagainwith Figure6 and supposeagain that E is the solutionchosenby the rule. Now alter the bargainingset in the following way: leave the "bestpossible" outcomes for the vertical andhorizontalpartiesunchanged,but fix thingsso that for each possible benefit to the hori-zontal partythe largest possiblegain to theverticalparty is bigger than it was before.Then the axiom of monotonicity requiresthatthe ruleassignto this newbargaining eta solution that gives the verticalpartymorethanat E. If the environment ecomesmorefavorable or the verticalparty n this strongsense, the verticalpartymustprofitfromthechange.(Of course the environment an be-come more favorable or both parties;thenthey mustboth gain.)Kalai and Smorodinsky how that replac-ing the axiom of irrelevance f independentalternativeswith the axiom of monotonicityleads to a unique solution different fromNash's. It is easily described.Let G* be thebest that the verticalparty could hope for,R(L)-wL in Figure 6. Let V* be the bestthehorizontalpartycould hope for.Find thepoint (G*, V*)and draw a line connecting twith theno-bargainpoint,whosecoordinatesin Figure6 are (0, NU(W)).The solution isthe uniquepoint at which that line intersectsthe efficiencyfrontier.It is shown in Figure6 as F.Although hegeometrys simple,the arith-metic of the Kalai-Smorodinskyolution isnot. The equation to be adjoinedto that ofthe contractcurve s(9) (R(L)-wL)/(L(U(w)-U(w-)))

w (R(L )-WaL)d(L(U()t)-U(h )))-where (w-,L) and (w,vL) are the left- and

right-hand ndpositionof thecontractcurve.Thus R'(L)= w-and R(L)m=iL. (We are as-suming here that L

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VOL. 71 NO. 5 McDONA LD AND SOLO W: WAGE BARGA NING 907

LD

LLMOX~~maxFIGURE 7

at Lmax' becausesuch pointsare inaccessible.In fact, thecontractcurvecoincideswith theverticalabove the intersectionpoint, as inFigure8. (Below the intersection,points onthe verticalare dominatedby points on thecontract curve to the left of LmaX.)A simplerecession story might go as fol-lows. To begin, the sales constraintis notbindingand thewage-employmentargain ssomewhere n the contractcurve.It remainstherein the early stagesof the recessionasLmax movesto theleft but is not yet binding.Eventually,the constraint just binds, andthen movesstill further o theleft. The initialbargain is no longer tenable. (We ignorelaborhoardingonly in order to concentrateon the logic of bargaining.)What happensnow?Givenanaccepted quityrule whichmightalso be shifting systematicallyas the re-cession proceeds)a new wage-employmentbargainmightbe struckat the intersection fthe recession-shiftedequity and contractcurves.Butsupposethe initialbargainA hadarisenmostlyby historicalaccident.It mightnot even be efficient.A natural ncrementalequityrulemightbe thatbothpartiesshouldgain,or bothlose, by thechange,but not onegainand theother lose. In the recessioncase,both must lose. This suggeststhat the newbargain would have L=Lm.ax and a wagesomewhere n the intervalBC between theisoprofit curve and the indifference curve

w New ControctCurve

OldC\I,c ControctCurveB A

w~~~~~~~~~~~~Lmox

FiGruRE8

associatedwith the initial bargain.This isshown n Figure8.Any point on BC has a wage higherthanA; the wagewould rise as employment alls.This is too sharpto be taken literally.Thelogic of this resultprovides,however,a clearinsight into the mechanismthroughwhichefficient bargaining can generate counter-cyclicalwagevariation.At any point on thecontractcurve,the firm wouldprefera lowervolume of employment at the bargainedwage. We have already suggestedthat con-tractualworkrulesand manningagreementsmight serve the purpose of enforcing thisextra employmenton the firm. A bindingsales constraintthus benefits the employerby necessitating, r providing he excusefor,a reduction n employment.At the old wage,the firm would be betteroff and the unionworse off. If the incremental quityrule for-bids such an outcome,the wagemustrisetotransfer omeof the loss from unionto firm.This showsup withgreatclarityherebecausethe recession s assumed o leave the revenuefunction unchangedexcept by imposing abarrier at Lmax.

VII. Conclusion

We set out to understandwhy real wagesmight be sticky, why fluctuations n aggre-

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908 THE AMERICAN ECONOMIC REVIEW DECEMBER 1981

gate demandmight have their major effecton employmentand little or none on thewage. Our partial-equilibriumbargainingmodels can hardly be expected to do that.But they do quite generallyconfirma ten-dency for fluctuations n real product de-mand at the firm or industry level to beaccompanied y large correlated luctuationsin employmentand small changes in realwage ratesthat could go in eitherdirection.What s the sourceof that tendency?Geometricallypeaking, t is becausebothour efficiency locus and our "equity" or"power"ocus shifts to the left in recessionand to the rightin upswings,providedcycli-cal changes in product markets dominatethose in the effectivereservationwage.Theshifts operatein the same direction on em-ployment,but in opposite directionson thenegotiatedwage, so there is a clearpossibil-ity that the two will be statistically ndepen-dent, as empiricalnvestigation uggests.A deeper, less mechanical,answer mightgo something ike this. Efficientbargainingpushes the firmto hire moreworkers hanitwould like at the negotiatedwage. The out-come is thus on the falling part of the iso-profit curve. The contract curve slopes up-ward.Higheremploymentand higherwagesfavor the workers;lower employment andlower wages favor the firm. When circum-stancesenforce a reduction n employment,the employer gains and the workerslose.Equity and bargainingpower are likely toseek an adjustment hat will transfer ome ofthe employer'sgain to the union or shiftsome of the union's loss back to the firm.The part of this adjustment hat falls (effi-ciently)on wageswill involvean increase nthe wage. This tendency can, in principle,offset thenormalcyclicaldeterioration f thedemand for labor,in part,wholly,or not atall.Most of the paperrepresents ariationsonthis theme. If short-term ontractingwouldlead to cyclicalfluctuation n employmentata more or less stablewage,thenconveniencecouldeasily lead the parties o contract or a

long-term teadywage, withcurrentemploy-mentdecisionsmade by the firm. The unionwould need protectionagainstexcessive i.e.,profitablebut "inefficient") eduction n theaverage evel of employment; his could beprovided by work rules or manning agree-ments.Ourmain result s in sharpcontrastto theoutcome of standardmodelsof implicit con-tractingwith symmetric nformation.There,long-runcontracts tend to be employmentstabilizing scomparedwithspot-competitivelabormarkets.Thecrucialdifferenceappearsto be thatimplicit-contractmodels areclosedby a utilityconstraint.We replace his condi-tion by the sort of equity convention thatarises naturally in the bargaining contextand is less dominatedby opportunities vail-able elsewheren the economy.

REFERENCESC. Azariadis, "Implicit Contracts and RelatedTopics: A Survey," unpublished workingpaper, 1979.A. M. Cartter, Theory of Wages and Employ-ment, Homewood: Irwin, 1959.GeorgedeMenil, Bargaining: Monopoly Powerversus Union Power, Cambridge, Mass.1971.P. T. GearyandJ. Kennan,"The Employment-Real Wage Relationship: An InternationalStudy," J. Polit. Econ., forthcoming.R. E. Hall andD. M. Lilien, "Efficient WageBargains under Uncertain Supply and De-mand," Amer. Econ. Rev., Dec. 1979, 69,868-79.E. KalaiandM. Smorodinsky,"OtherSolutionsto Nash's Bargaining Problem," Econo-metrica, May 1975, 43, 513-18.W. Leontief, "The Pure Theory of theGuaranteed Annual Wage Contract," J.Polit. Econ., Feb. 1946, 54, 76-79.H. Raiffa, "Arbitration Schemes for Gener-alized Two-Person Games," in H. W. Kuhnand A. W. Tucker, Contributions To TheTheoryof Games, II, Princeton 1953, 361-87.