Efficient wage bargaining in a dynamic macroeconomic model

Institute of Mathematical Economics Working Papers

465

March 2012

Efficient Wage Bargaining in a Dynamic Macroeconomic Model

Volker B¨ ohm and Oliver Claas

IMW · Bielefeld University

Postfach 100131

33501 Bielefeld · Germany

email: imw@wiwi.uni-bielefeld.de

Dynami Maroeonomi Model

Volker Böhm

∗

Oliver Claas

†

Marh 26, 2012

Abstrat

Thispaperanalyzestheimpliationsofbilateralbargainingoverwagesandemployment

between a produer and a union representing a nite number of idential workers in a

monetarymaroeonomimodeloftheASADtypewithgovernmentativity. Wagesand

aggregateemploymentlevelsaresetaordingtoaneient (Nash)bargainingagreement

whiletheommoditymarketislearedinaompetitiveway. Itisshownthat,foreahlevel

ofunionpower, measuredbytheshare itobtainsof thetotalprodution surplus,eient

bargainingimpliesnoeieny lossinprodution. Dependingonthelevelofunionpower,

temporaryequilibriamayexhibitvoluntaryoveremploymentorunderemploymentwiththe

ompetitive equilibriumbeinga speialase.

Duetothe priefeedbakfromtheommoditymarket andto inome-indueddemand

eets,alltemporaryequilibriawithapositivelaborsharearenotNashbargaining-eient

withrespetto theset of feasibletemporary equilibrium alloations. Whilehigher union

powerinduesalargershareofthesurplusandahigherrealwage,italwaysimplieslower

output and employment. Moreover, the indued nominalequilibrium wage is not always

a monotonially inreasing funtion of union power. Therefore, all temporary equilibria

witheient bargainingareonlySeond-best Paretooptimal, i.e.bargainingpowerand

prodution eieny donot lead to temporary optimality.

The dynamievolutionofmoney balanes, pries, andwagesisanalyzed being driven

primarily bygovernment budgetdeitsandexpetations byonsumers. It isshownthat

for eah xed level of union power, the features of the dynamis under perfet foresight

arestruturallyidentialto thoseof thesame eonomy underompetitive wage and prie

setting. These are: stationary equilibria with perfet foresight do not exist, exept on a

set of parameters of measure zero; balaned paths of monetary expansion or ontration

are the only possibilities induing onstant alloations; for small levels of government

demand, there exist two balaned paths generially, one of whih with high employment

and produtionis always unstable,while theotherone maybe stableor unstable.

Keywords: Eient Colletive Bargaining, Union Power, Monopolisti Wage Determination,

Aggregate DemandAggregate Supply, Employment, Pries, Wages, Ination, Expetations,

Government Deits, Monetary Expansion,Perfet Foresight, Dynamis, Stability

JEL Classiation: C78, D33, D42, D43,D58, D61, E24, E25, E31, E41, E42, J42,J52

∗

DepartmentofEonomis, BielefeldUniversity,vboehmwiwi.uni-bielefeld.de

†

InstituteofMathematialEonomis(IMW)andBielefeldGraduateShoolofEonomisandManagement

(BiGSEM), BielefeldUniversity, oliver.laasuni-bielefeld.de. Thisresearh wasarriedoutwithin the

InternationalResearh Training GroupEonomi Behaviorand Interation Models (EBIM) naned bythe

GermanResearhFoundation(DFG) underontratGRK1134/2.

Contents

1 Introdution 3

2 The Labor Market with Eient Bargaining 5

2.1 The Publi Setor . . . 5

2.2 The ProdutionSetor . . . 6

2.3 The ConsumptionSetor . . . 6

2.4 Eient Wage Bargainingand Employment . . . 8

2.5 NonompetitiveWage Setting versus Wage Bargaining . . . 14

3 Temporary Equilibrium with Eient Wage Bargaining 16 3.1 The Roleof Union Power inTemporaryEquilibrium. . . 16

3.2 ComparingBargaining and Competition . . . 22

3.3 Ineient Redistributionunder Eient Wage Bargaining . . . 25

3.4 Summary . . . 27

4 A Parametri Example: the Isoelasti Case 28 4.1 The Roleof Union Power . . . 31

4.2 Union Powerand Wages . . . 33

4.3 ComparingWages, Pries, and Payos . . . 37

5 Dynamis of Monetary Equilibrium 41 5.1 Perfet Foresight . . . 41

5.2 Dynamis of Money Balanes and Pries . . . 43

5.3 Steady States and Stability . . . 44

5.4 Dynamis of RealMoney Balanes under Perfet Foresight . . . 47

5.5 StableBalaned Paths . . . 49

6 Summary and Conlusion 51

Bibliography 52

1 Introdution

In spite of the fat that in most industrialized ountries negotiations between workers unions

andsyndiatesofproduersaboutwage levelsandemploymentonditionsourregularly,their

eonomisignianeforthe labormarketorevenmore fortheevolutionof themaroeonomy

as awhole is oftennegleted inthe researhon labormarkets.

1

Taking the number of artiles

on the subjet in the reent Handbook of Labor Eonomis by Ashenfelter & Card (2011a, b)

relativetootherontributionsthereinasanindiator,itseemsthatothertheoriesareonsidered

asmorerelevantandthemotivationtostudytheimpatofbargainingbetweenthetwosideson

apartiularmarketarenotattheforefrontoftheresearhinlaboreonomis. Amongthemany

possiblemaroeonomimodelswhihdeterminewageandemploymentlevels,thosewhihtake

a bargaining approah between a produers onglomerate and a workers union are learly in

the minority. This is in ontrast to the general empirial observation that suh negotiations

areobservablereurringannualeventsinmostWesterneonomieswhihinduelegallybinding

agreementswhih are adhered to inthese eonomies.

Considering the theoretial models of bargaining between groups (as opposed to other wage-

employment-determining proedures) 2

from a general miroeonomi perspetive, the impor-

tane of strategi aspets in wage and employment negotiations are well reognized and have

been studiedextensively. The literatureontainsseveral ontributionsapplyinggame-theoreti

notions and onepts (see for example MDonald & Solow 1981; Landmann & Jerger 1999;

Gerber&Upmann2006). However, mostofthem ignoreross-marketeets andarry out the

analysis in a partial-equilibriumsetting. Thus, any spillovers from other markets or from the

inome distribution onthe general-equilibriumor maroeonomi level are rarely disussed or

analyzed, whih redues the validityof their results as ontributions tomaroeonomis.

Oneexplanationforthelakofmoreextended game-theoretionsiderationsinmaroeonomi

modelsmaylieinthelimitationsofthegame-theoretiapproahesandtheirmodelsthemselves.

Two essential aspets may explain this absene:

1. the interation of the labormarketwith the rest of the eonomy, and

2. the dynami aspet of reurring negotiations, of time, and of unertainty.

Withrespettotherstpoint,theexistingtheoriesarebuiltprimarilyontheommonpriniple

of bargainingas analloationdevie ofhow todivide aake of given size. Ifthere were strong

empirialevideneoraonvining theoretialargumentthat infatinmost marketeonomies

thelabormarketisasuientlyindependentandisolatedunitwithintheeonomy,whoserules

and alloation priniples have little inuene on the size of the ake, i.e. on GNP, then the

underlying premise of a given onstant ake would be justied, and the distributive aspets

ould be separated from the alloativeissues on the national level. However, most eonomists

would agree that there are major alloative mehanisms originating from labor market rules

to the maroeonomi level. Suh spillovers or feedbak eets play a role in determining

the size of GNP. In addition, most game theorists would also agree that many appliations

of bargaining theories assume too naively that the negotiations are direted toward outomes

to be distributed. In most situations, however, bargaining agreements onsist of priniples

or rules in an alloative environment. Outomes are the onsequenes after the behavioral

1

Inontrast,thesoialandlegalaspetsofwageontrats,of hiringand ringaredisussed andanalyzed

toalargedegree.

2

suh aseienywages,searhtheory,mathing theory,et.

responseof agentsarossmarkets. In otherwords,outomes resultafterthe feedbaks between

markets take plaeand the naloutome likeGNPand itsdistributive partsare endogenously

determined.

3

There are always behavioral responses originating from demand and supply behavior, from

outside options, and in partiular from the feedbak eets from other markets and through

inome eets. Thus, maroeonomi outomes are the result after behavioral onsequenes

in the markets and the spillovers indued, implying that the size of the ake depends on the

rules set in the negotiations. Therefore, muh of standard bargaining theory may not even

be appliable in suh ases or has to be reevaluated. It provides essentially a stati solution

onept and frameworkfor negotiations with noonsiderationfor interation orfeedbak with

an environment ormodel. Considerations for impliations for outomes afterindued hanges

of the environment inludingthe feedbak are absent.

For the dynami impliations of repeated negotiations ourring in maroeonomi systems,

game theory again does not provide modeling approahes at a satisfatory level to be applied

suitably tolabor markets. The issues tobe solved in asetting of repeated negotiations open a

wide range ofunsolved problems astothe dynami setting ofthe negotiation, the negotiators,

the environment, the state variables,and the information,unertainty, and stohasti shoks.

Again, with the ross-market feedbaks playing a qualitative role, the negotiations and their

proedures will have aninuene onthe dynami evolution of the eonomy.

The literature on the usage of eient bargaining taking a maroeonomi perspetive is not

sizable.

4

MDonald & Solow (1981) study nonompetitive wage setting in partial equilibrium

models with apaity-onstrained,fully unionized labormarkets with one rm and one union.

Interalia,theyanalyzetheasesofthemonopolistiunion(withtherighttomanageoftherm)

as well as two types of eient bargaining over wages and employment using the symmetri

Nashresp.theKalaiSmorodinskybargainingsolutions. Theagents'objetivefuntionsarethe

protof the rm resp.the expeted exess indiretutilityof the representativeunion member.

Indiret utility ismeasured in nominalwages for a onstant reservation wage.

5

Booth (1996)and Landmann &Jerger (1999) are two prominentpresentations addressing and

disussing the eient bargaining solution expliitly in a format whih is the losest to the

one proposed here. Booth (1996) slightlyextends the settingbyMDonald & Solow(1981)by

applyingthegeneralizedNashbargainingsolutionwhileanalyzingbargainingoverwagesalone.

This leaves the employment deision to the rm whih orresponds to the so alled right-to-

manage model. Her modeling generalizes the monopolisti-union model and shows that the

resultingoutome isnot Pareto eient ina stati partial-equilibriumsetting.

Landmann &Jerger(1999) presentthe eientbargainingmodelwhere intertemporalaspets

or money plays no role. They present a partial-equilibrium analysis only by assuming xed

3

There are many examples from empirial agreements whih onrm this fat. For example wage laws

for union members, indexed wages rules, minimum wage laws. Trade agreements among ountries speify

priniples of afree trade: notaris orduties, no disrimination rules, harmonizationof taxes, asin theEU.

Finanial/monetarypriniplesinamonetaryunionspeifyaommonurreny,mutualfreeexhange,likeIMF,

ECB.Cartelagreementsspeifyrulespresribingdos anddon'ts.

4

Wearenotawareofanypubliationsanalyzingtheroleofeientbargainingandspilloversarossmarkets

norofthedynamis inalosedmaromodel.

5

There are some ontributions dealingwith spei dynami orpoliy issues within models of apital a-

umulation, as for example Devereux & Lokwood (1991);Kaas & von Thadden (2004);Gerber & Upmann

(2006);Koskela&Puhakka(2006)withinnonmonetarymodels. Gertler&Trigari(2009)presentsaninteresting

ombinationofamarketwithmathingand staggeredNashbargaininginanempiriallyorientedmodel.

pries throughout with no analysis of the demand side of the eonomy orthe eets from the

inome distribution. Moreover, noomparative statisanalysis of the role of union power and

their impliations foralloationsisperformed.

This paper starts from the general premise that there are signiant feedbaks to be studied,

whih are shown to exist in the standard ASAD model of a monetary maroeonomy. It

analyzes the most innouous so-alled eient bargaining solution for the labor market as a

benhmark model, whih assumesthe most ooperativestruture and solutiononept froma

strategipointofview. Whilethe literatureagreesthat thissolutiononept isempiriallythe

most unlikely, itsimpliations forthe maroeonomymust beexamined,in partiularwhether

it indues the qualitativeproperties of eieny and optimality whih the literature seems to

assign toit.

The paperderivesthe struture ofthe temporarypriefeedbak and disusses thefullompar-

ativestatisof varyingunionpower, indiatingthat,inspite oftheappliationof theeieny

riterionusedinthelabormarketseparately,theeienyriterionaswellasParetooptimality

fails on the marolevel. It ompares the alloative onsequenes with other strategisolutions

of nonooperativebehavior of produersand the union. Finally,the dynamionsequenes for

alloationsand the stabilityof the evolutionunder perfet foresight are investigated.

2 The Labor Market with Eient Bargaining

Consideraneonomyindisretetimewiththreemarkets: alabormarket,aommoditymarket,

and a money market, and three setors: a onsumption setor, a prodution setor, and the

publi setoronsisting of aentralgovernment and a entral bank.

6

2.1 The Publi Setor

The publi setor onsists of a government and a entral bank. The government demands

the produed ommodity at a level

g ≥ 0

^{to produe publi goods and servies. These are}

assumed tobe pure publigoodsprovidinga onstant level of utilityeah periodto eahtype

of onsumer. In addition, onsumer preferenes are assumed to be additively separable with

respettothelevelofthepubligoodsothatthesedonotinduemarginalorbehavioraleets

by onsumers. Therefore, the onstant level of publi servies an be and was dropped as an

argument inonsumer utility funtions.

Tonaneitsonsumption(thepubligood'sprodution)thegovernmentleviesaproportional

taxonprotsattherate

0 ≤ τ _π ≤ 1

^{andonwagesattherate}

0 ≤ τ _w ≤ 1

^{. Sinethegovernment}

parameters are assumed tobe given parametriallyin eahperiod,in general,the government

budget is not balaned sine inomes are endogenously determined. Therefore, the entral

bank reates/destroys the amount of money aording to the need of the government arising

from the unbalaned budget. Sine money is the only intertemporal store of value held by

onsumers, any inrease (derease) of the amount of money required to balane the budget of

6

Themodelhosenis astandardversionof an ASAD model basedon miroeonomipriniples and em-

beddedinaneonomywithohortsofoverlappinggenerationsofonsumers(seeforexampleBöhm2010).

the government is equivalent to the amountof savings (hanges of the amount of money held

by the private setor)in any given period.

7

2.2 The Prodution Setor

The nonstorable ommodity is produed from labor only by a single prot-maximizing rm.

8

The tehnology of the single produing rm is desribed by a dierentiable monotonially

inreasing and onave prodution

F : R → R

^,

L 7→ F (L)

^satisfying

F (0) = 0

^{and the usual}

Inada onditionswhihimplies thatthe tehnialequipmentorthe stok of apitalisonstant

and does not depreiate.

At a given nominal wage rate

w ≥ 0

^{for labor and a sales prie}

p ≥ 0

^{for the ommodity, a}

prodution deision

L

^{implies urrent prots}

Π(p, w, L) := pF (L) − wL

^{. All prots are paid}

to onsumers, who are the owners or the shareholders of the rm. There is no intertemporal

deision making of the produer with noneed to retain prots nor tohold money. Therefore,

therm'sobjetiveistomaximizeprots. Under ompetitiveonditionswithpriesandwages

given, the behavior of the rm in eah period in the two markets indues the usual prot-

maximizinglabordemand funtion

h

^om

w

p

:= arg max

L ≥ 0 {pF (L) − wL} = (F ^′ ) ^{− 1} w

p

(1)

and the ommodity supply funtion

F (h

^om

(w/p))

^.

In nonompetitivesituations, in partiular under bargaining, pairs

(L, w)

^{of employment and}

wage levels have to guarantee nonnegative prots

Π(p, w, L) ≥ 0

^{for the produer. Therefore,}

the zero-prot ontour implies the partiipationonstraint for the produer

w ≤ p F (L)

L =: W Π (p, L),

whihdenes hisreservation wage as afuntion of the employmentlevel

L > 0

^.

2.3 The Consumption Setor

The onsumption setor onsists of overlapping generations of two types of homogeneous on-

sumers. There are

n _w ≥ 1

^{workers and}

n _s ≥ 1

shareholders ineah generation, both of whih livefortwoonseutiveperiods. Thesizeandompositionofthetwogroupsisonstantthrough

time implying that atany one time,there are

n s + n w

^{young resp. old onsumers.}

Eah shareholder onsumer reeives net prots only in the rst period of his life. He spends

the proportion

0 < c(θ ^e ) < 1

^{in the rst period and saves the rest inthe form of money to be}

spent ononsumptioninthe seondperiod. Money isthe only intertemporalstoreof value for

7

Tosaveonnotation,weomit,whereverpossible,thegovernmentparameters

g

^,

τ w

^,and

τ π

^{inallarguments}

throughout this paper. Whenanalyzing behaviorand markets in any partiularperiod, it isalwaysassumed

that moneyholdings

M ≥ 0

^andprieexpetations

p ^e > 0

^{aregivenat thebeginningandremainxed during}

theperiod,exeptwhentheiromparativestatiseets aredisussed.

8

This assumption ismade for simpliityonly, the extensionto multiple homogeneousrmsorganizedin a

produersassoiationisstraightforward.

onsumers whih arries no interest. Therefore, his onsumption/savings deision depends on

the expeted rate of ination

θ ^e := p ^e /p

^.

Eahworkersupplieslaborintherstperiodofhislifetoonsumeintheseondperiodonly. His

preferenes withrespet toplanned future onsumption

c ^e ≥ 0

^andwork

ℓ ≥ 0

^whenyoungare

desribedbyanintertemporalutilityfuntionoftheform

u(ℓ, c ^e ) = c ^e −v (ℓ)

^{wherethefuntion}

v : R + → R +

^{measuresthe disutilityfromlabor. The funtion}

v

^{isassumed tobe} ontinuously dierentiable, stritly monotonially inreasing, stritly onvex, with

v(0) = v ^′ (0) = 0

^and

lim ℓ→∞ v ^′ (ℓ) = ∞

^.

Given a wage rate

w

^{, an employment level}

ℓ

^{, and a wage tax}

τ w

^{, he saves his total nominal}

net wage inome

(1 − τ _w )wℓ

^{in the formof money, to be spent on onsumptionin the seond}

period of his life. With given prie expetations

p ^e

^{, his planned future onsumption satises}

p ^e c ^e = (1 − τ w )wℓ

^{. Therefore, under ompetitive onditions and prie} expetations

p ^e

^{, his}

utility-maximizinglaborsupply is given by

arg max

ℓ≥0

u

ℓ, (1 − τ w ) w p ^e ℓ

= (v ^′ ) ^{− 1}

(1 − τ w ) w p ^e

,

whih is a ontinuous, stritly monotonially inreasing funtion of the expeted future value

of the urrent nominal wage.

Given the worker's prie expetations

p ^e > 0

^{, it is} straightforward to dene his reservation wage for nonompetitive situations. The labormarket partiipationonstraint of a worker for

an aeptable employmentwage situation

(ℓ, w)

^{must provide a utility atleast ashigh asnot}

working when young. In otherwords,

(ℓ, w)

^{must be asolution of}

u(0, 0) = 0 ≤ u(ℓ, c ^e ) = u

ℓ, (1 − τ w ) w p ^e ℓ

= (1 − τ w ) w

p ^e ℓ − v(ℓ).

Thisimpliesthelowerboundoftheindividuallyaeptablewagerate,i.e.hisreservationwage,

as

w

p ^e = 1 1 − τ w

v (ℓ)

ℓ , ℓ > 0

⁽²⁾

whih is a stritly inreasing funtion of the employment level. >From these properties one

denes diretly the aggregate ompetitive laborsupply as

N

^om

w

p ^e

:= n _w ℓ = n _w (v ^′ ) ⁻¹

(1 − τ _w ) w p ^e

whihhas a global inverse given by

w

p ^e = S

^om

(L) := 1 1 − τ w

v ^′ L

n w

.

Withequaltreatmentofworkers one obtainsthe aggregatereservationwage fromequation (2)

as

w

p ^e = S(L) := n w

L(1 − τ w ) v L

n w

,

whihhas an elastiity 9

E _S (L) = E _v (L/n _w ) − 1.

⁽³⁾

9

For any funtion

f

^{we denote itselastiity at}

x

^as

E f (x)

^{. Thus,}

E v (L/n w )

^{denotes the elastiity of the}

funtion

v

^.

This implies a useful relationship between the reservation wage and the ompetitive inverse

laborsupply funtion

S

^om

(L) = E v (L/n w ) S(L)

^forall

L.

⁽⁴⁾

Given the harateristis of eah individual young worker, the union is pereived of as anag-

gregateagent representing the onsumer-workers onsisting of allhomogeneousworkers. Sine

all workers have idential harateristis, the union's bargaining will be onerned with the

determinationof the wage level

w

^{and the aggregate level of employment}

L

^{, assumingthat all}

workers aretreatedequally,i.e.eahispaidthewage

w

^{withindividualemploymentlevel}

L/n w

^.

2.4 Eient Wage Bargaining and Employment

It is evident that one of the most hallenging questions to investigate onerns the feedbak

eets or spillovereets between the labormarket and the output market sine in the losed

maroeonomythe impatfrom wage negotiations onthe inomedistribution willhaveeets

onaggregate demandand thereforeonoutputand inome. Moreover, theseeets willdepend

onthe marketstruture hosenon either side.

The framework hosen for the wage bargaining between the union representing the onsumer-

workersandtheproduerasawagedeterminationdevieonsistsofanappliationofabargain-

ingsolutiontothesimultaneous determinationofthe aggregateemploymentlevel

L

^{and of the}

wagerate

w

^{undertheassumptionthatthe} negotiatingparties,theunionandtheproduer,are both prie takers inthe ommodity market. Withthis hoie it is possible to disuss best the

role ofbargaining ingeneralequilibriumandompare the outomes withthe ompetitivease.

Under eieny onsiderations, hoosingthe Nashbargaining solutionouldbeone possibility

although inthe repeated or dynami ontext this may not be the fully onvining.

10

In other

words, the produer and the union treat the ommodity prie as given, impliitly assuming

that their bargaining deision has no inuene on the indued equilibrium prie in the short

run. Thus, a temporary equilibrium with eient wage bargaining is dened by a ompetitive

prielevel

p

^{whihequalizesaggregate supplyand aggregatedemand ofthe ommoditymarket}

at whih the levels of employment and wages indue the desired eient bargaining solution

between the union and the produer.

The result of the bargaining proedure between the unionand the produeronsists of a joint

deision with respet to the employment level

L

^{and the wage rate}

w

^{where the produer's}

goalis to maximizeits net prot while the union tries to maximizethe aggregate exess wage

bill for the workers. Let

Π(p, w, L) = pF (L) − wL

^{denote the net prot and}

Ω(p ^e , w, L) :=

wL − p ^e S(L)L

^{theexess wage bill. Given prie}expetations and ommodityprie

(p ^e , p) ≫ 0

^,

a bargaining agreement

(L, w)

^{is alled} individually rational if

Π

^and

Ω

^are nonnegative. An eient bargaining agreement between the unionand theemployeris denedinthe usual way.

Denition 2.1 Given

(p ^e , p) ≫ 0

^{, a} employmentwage pair

(L, w) ∈ R ² +

^{is alled eient if}

there exists no other pair

(L ^′ , w ^′ )

^suhthat

Π(p, w ^′ , L ^′ ) ≥ Π(p, w, L)

^and

Ω(p ^e , w ^′ , L ^′ ) ≥ Ω(p ^e , w, L)

with at least one strit inequality.

10

Fromagame-theoretipointofview,thegeneralizedZeuthensolutionforhalf-spaegamesanbeapplied

whih islessspeithanNash;seealsotheremarksintheintrodutionandintheonlusion.

To haraterize eient agreements, one may use the assoiatedLagrangean funtion

Λ(w, L, κ) = Ω(p ^e , w, L) + κ Π(p, w, L) − Π ¯

and obtains the rst-order onditions of aninteriorsolution

(L, w) ≫ 0

^as

pF ^′ (L) = p ^e (S(L) + S ^′ (L)L), L > 0.

⁽⁵⁾

Any positive solution determines the same level of employment for all levels of net prot

Π ¯

^.

Moreover, the solution of (5) isidential with that level of employment whih would lear the

labor marketunder onditions of perfet ompetition between the union and the produer for

any given pair

(p ^e , p) ≫ 0

^.

This result iswell-known fromthe literature. It ours insituations ofbargaining/ooperative

deision making between any two agents who are the only partiipants trading in the same

market, whih orresponds to the situation in a vertially integrated industry, a artel or a

bilateralmonopoly. In suhases, undereieny, the two tradersinternalizeall potential net

gainsand theywilldeideonaleveloftradeandpriebetweenthemwhihmaximizesthe sum

of their net gains. Ifthey are both faingompetitivemarkets upstream and downstream, the

resultinglevelofativity between themundereieny isidentialtothat leveloftrade whih

wouldresultunderompetitivetrading,withsomemildassumptions. Thislevelguaranteesthat

therearenofurtherjointgainstoshare. Inotherwords,theleveloftradeequalizesmarginalost

to marginalrevenue between the two players and maximizes the ake toshare. Forthe model

here between the union and the produer, this implies that the determination of an eient

bargaining solutionan bedivided intotwosteps: the hoie ofthe levelof employment whih

depends on the market data upstream and downstream, and the determination of the wage

whih then turns out to beome the entral point in the bargaining proedure of sharing the

net gains.

Wage Bargaining in the Bilateral Monopoly

As pointed out in the previous paragraph, the employmentdeision under eient bargaining

turns out to be equivalent to the standard textbook representation when the union and the

produer form a bilateral monopoly. For a given prie expetations and ommodity prie

(p ^e , p) ≫ 0

^{,the jointnet gain isgiven by}

Π(p, w, L) + Ω(p ^e , w, L) = pF (L) − wL + wL − p ^e S(L)L = pF (L) − p ^e S(L)L

is afuntion ofthe employmentlevelalone. Thus, itis neessary that an optimalemployment

deision maximizes

pF (L) − p ^e S(L)L

^, independent of the wage deision to be taken. This indues the rst-order ondition

pF ^′ (L) = p ^e S(L) E S (L) + 1

⁽³⁾

= p ^e S(L) E v (L/n w ) − 1 + 1

⁽⁴⁾

= p ^e S

^om

(L),

⁽⁶⁾

whih oinides with (5). Therefore, the employment deision of a bilateral monopoly max-

imizing joint net gain against the rest of the eonomy oinides with the one under eient

bargaining. Thus, the employment deision to yield the maximal joint net gain an be sepa-

ratedfromthe wagedeisionofhowthisgain istobedistributed. Inthisperspetive,thelabor

market has been eliminated, the employment deision

L

^{orresponds to an internal deision}

of a union-produermonopoly, while the deision forthe wage rate beomesa ost alloation

issue.

This separability of the employment and the wage deision an be portrayed geometrially in

the assoiatedemploymentwage spae (see Figure1). For

L > 0

^{,anaeptable wage must be}

suh that

Π ≥ 0

^and

Ω ≥ 0

^{, i.e.}

w ≤ p F (L)

L = W Π (p, L)

^and

w ≥ p ^e S(L) =: W Ω (p ^e , L),

induing the two status-quo wage funtions

W Π

^and

W Ω

^{whih orrespond to the} reservation wage of the produer and of the union respetively. The area between the two funtions in

Figure1 denes the set of individually rationalemploymentwage pairs.

The set ofeientemploymentwagehoiesunderbargaining arethose onthe ontrat urve

shown as the bold red line. Geometrially speaking, eah point on the ontrat urve must

PSfragreplaements

0

0 L

w

W Π

W Ω

(W Π L) ^′ (W Ω L) ^′

Figure1: Determiningthe level of employment

be a tangeny point of an iso-utility and of an iso-prot urve (the thin lines). Sine all iso-

utility/iso-proturves are of the form

W Π ¯ (L) = pF (L) − Π ¯

L

^resp.

W Ω ¯ (L) = p ^e S(L) + Ω ¯ L

for alllevels

Π ¯

^and

Ω ¯

^{, the tangeny ondition}

∂W (L)/∂L

^implies

pF ^′ (L)L − W (L)L

L ²

= ! p ^e S ^′ (L) − W (L) − p ^e S(L)

L .

Sine

F (L)

^and

−S(L)L

^{are stritly onave funtionssatisfying the Inada onditions, the set}

of individually rational

(L, w)

^{is ompat. Moreover,}

pF (L) − p ^e S(L)L

^{is a stritly onave}

funtion as well. Therefore, the neessary onditions are also suient. Finally, given the

strit onavity ofboth funtions,the solution

L > 0

^{is unique forany positivegiven expeted}

inationrate

θ ^e = p ^e /p > 0

^{. Thus, the solutionofequation(5)denes anemploymentfuntion}

h : R ++ → R ++

^,

θ ^e 7→ h(θ ^e )

^{. Itsinverse isgiven expliitlyby}

p ^e

p = F ^′ (L) S(L) + S ^′ (L)L

(6)

= F ^′ (L)

S

^om

(L) := h ⁻¹ (L),

⁽⁷⁾

whih is dierentiable and stritly dereasing sine

(h ^{− 1} ) ^′ (L) < 0

^{holds. Therefore, under}

eient bargaining, the level of employment

h(θ ^e )

^{is a} well-dened, stritly monotonially dereasing, and invertible funtion of the expeted ination rate

θ ^e

^{. It is} homogeneous of degree zero in prie expetations and pries, it is dereasing in expeted pries and inreasing

in the urrent output prie. In addition, the employment level hosen by the two bargaining

parties isthe same as the one whihwould result in equilibriumundera perfetly ompetitive

labormarket.

Rewritingthe ondition (7)using the two reservation wage funtions, one obtainsan intuitive

and interesting relationship

W _Ω (p ^e , L) = p ^e S(L) = E F (L) E S (L) + 1

pF (L)

L = E F (L)

E S (L) + 1 W _Π (p, L).

⁽⁸⁾

for the relative shares depending on the elastiities of the reservation wage funtions, whih

also haraterizes the bargaining level of employment. This stipulates that the ratio between

the two status-quovalues should orrespond to the ratio of their respetive elastiities.

The Wage Rate under Bargaining

Given

(p ^e , p) ≫ 0

^and

L = h(p ^e /p) > 0

^{, the bargaining deision between the two parties}

onerning the wage rate now onstitutes abargaining game with onstant transfers sine

Π + Ω = pF (L)−p ^e S(L)L = W Π (p, L)L−W Ω (p ^e , L)L

^{isaonstantsum. Thus, oneobtainsaspeial}

aseofabargainingproblem,towhihthegeneralizedZeuthensolutionapplies(seeRosenmüller

2000). Forsuhgames the bargaining power between the two partiesis usually measured by a

number

0 ≤ λ ≤ 1

^{, whihdenes the relative shareof the totalake tobe allotedto the party}

havingbargainingpower

λ

^{. Thus, foraonstanttotalgain}

Π + Ω = W Π (p, L)L − W Ω (p ^e , L)L

^,

the weights

(λ, 1 − λ)

^{determine a linear} redistribution of the total net gain among the two agents.

Therefore,with

L > 0

^and

0 ≤ λ ≤ 1

^{given,anappliationofthe}generalizedZeuthensolution 11

to the total gain implies hoosing the bargaining wage as a onvex ombination of the two

reservation wage levels

W Π

^(when

Π = 0

^)and

W Ω

^(when

Ω = 0

^{)with the same weights}

W (p ^e , λ, p, L) = λW _Π (p, L) + (1 − λ)W _Ω (p ^e , L), L = h(θ ^e ).

⁽⁹⁾

Substituting (9)into the utility and into the prot funtionsyields the payo vetor

(Π, Ω)

^of

the bargaining solution

Π(p ^e , λ, p, L) Ω(p ^e , λ, p, L)

!

= pF (L) − W (p ^e , λ, p, L)L W (p ^e , λ, p, L)L − p ^e S(L)L

!

= W _Π (p, L)L − W (p ^e , λ, p, L)L W (p ^e , λ, p, L)L − W Ω (p ^e , L)L

!

= W Π (p, L) − W Ω (p ^e , L)

L 1 − λ λ

!

= pF (L) − p ^e S(L)L 1 − λ λ

! .

(10)

For given

(p ^e , p)

^{, Figure 2 displays the range of the mapping (10) for dierent values of the}

parameter

λ

^{,revealingitslinearimpatonthe payo}distribution. Asimilarlinearrelationship 11

NotethatthegeneralizedZeuthensolution(whihanonlybeappliedtohalf-spaegames)oinideswith

thegeneralizedNashsolution,yetrequiringlessproperties.

PSfragreplaements

Π Ω

λ = 0.00 λ = 0.33

λ = 0.67 λ = 1.00

Figure2: The impatof the bargaining power

λ

^{on the} equilibriumpayo

holds for the role of

λ

^{on the bargaining wage. Finally,} substituting (8) into the bargaining wage funtion(9),one nds that the equilibriumbargaining wage

W (p ^e , λ, p, L) =

λ + (1 − λ) E F (L) E _S (L) + 1

pF (L) L

=

E F (L)

E S (L) + 1 + λ E S (L) + 1 − E F (L) E S (L) + 1

pF (L) L

is amultipleof average produtivity, and that the equilibriumreal wage

W (p ^e , λ, p, L)

p = 1

E F (L)

E _F (L)

E S (L) + 1 + λ E _S (L) + 1 − E _F (L) E S (L) + 1

F ^′ (L)

is a positive multiple of the marginal produt of labor (with

L = h(p ^e /p)

^{). Both equations}

showlearlyhowthe bargainingparameterinteratswith the elastiitiesofthe tworeservation

wage funtions

Relative Union Power

As was seen above, an eient bargaining solution

(L, w) = (h(p ^e /p), W (p ^e , λ, p, h(p ^e /p)))

^is

dened parametrially for a given

0 ≤ λ ≤ 1

^{measuring the bargaining power. Thus, the}

model does not provide a fully endogenous determination of the bargaining power between

the union and the produer. However, the eient level of employment is independent of

λ

^,

implying that unionemployernegotiations doguarantee produtive eieny. Therefore, the

bargaining parameter

λ

^{determines exlusively the} redistribution of revenue between the two parties, i.e. the share of wages and prots in totalrevenue.

It is intuitively lear (and also evident from the geometry of Figure 1) that there must be a

uniquebargaininglevelforwhihthe partiesagreeontheompetitivewage. This oneequalizes

marginal ost resp. marginal revenue (

(W Π L) ^′

^resp.

(W Ω L) ^′

^). Geometrially speaking, this

orresponds to the wage where the respetive iso-utility and iso-prot urves are horizontal.

Let the unique

λ

^{for whih this ondition holds be denoted by}

λ

^{nat, the natural}

λ

^{. It is the}

solutionof either

W (p ^e , λ, p, L) = ^! ∂(W Π (p, L)L)

∂L

^or

W (p ^e , λ, p, L) = ^! ∂(W Ω (p ^e , L)L)

∂L ,

where

L = h(p ^e /p)

^{. Insertingthe denition of}

W (p ^e , λ, p, L)

^{intothe seond equation gives}

λ

^nat

W Π (p, L) + (1 − λ

^nat

)W Ω (p ^e , L) = ∂(W _Π (p, L)L)

∂L = pF ^′ (L) = E F (L)W Π (p, L).

Exploiting (8)then gives

E F (L)W Π (p, L) = λ

^nat

W Π (p, L) + (1 − λ

^nat

)W Ω (p ^e , L)

= λ

^nat

W Π (p, L) + (1 − λ

^nat

) E F (L)

E S (L) + 1 W Π (p, L)

=

λ

^nat

+ (1 − λ

^nat

) E F (L) E S (L) + 1

W Π (p, L)

=

E F (L)

E S (L) + 1 + λ

^nat

E S (L) + 1 − E F (L) E S (L) + 1

W Π (p, L)

whihimplies

λ

^nat

(L) = E _F (L)E _S (L)

E S (L) + 1 − E F (L) .

⁽¹¹⁾

Inotherwords,

λ

^nat

(L)

^{isdeterminedby the}elastiities

E S

^and

E F

^{ofthelaborsupplyfuntion}

and of the prodution funtion respetively. Therefore, with isoelasti funtions

λ

^nat

(L)

^is

onstant.

The wage shareof total revenue an beomputed in asimilar manner.

wL

py = W (p ^e , λ, p, L)

W Π (p, L) = λ + (1 − λ) W Ω (p ^e , L) W Π (p, L)

(8)

= λ + (1 − λ) E F (L) E S (L) + 1

= E F (L) E S (L) + 1 + λ

1 − E F (L) E S (L) + 1

∈

E F (L) E S (L) + 1 , 1

.

(12)

Therefore, the prot shareof total revenue is

π

py = 1 − wL

py = (1 − λ)

1 − E F (L) E S (L) + 1

.

⁽¹³⁾

Notethatthewageshareresp.theprotsharefor

λ

^nat

(L)

^is

E _F (L)

^resp.

1−E _F (L)

^,asexpeted,

sine at

λ

^nat

(L)

^{the fator shares intotal output must be equaltothe respetive} elastiitiesof the produtionfuntion

F

^.

Underemployment and Overemployment

Sine the bargaining solution

(L, w) = (h(θ ^e ), W (p ^e , λ, p, h(θ ^e )))

^{is a joint agreement between}

the two agents, there an neither be any involuntary unemployment nor overemployment. In

other words, any dierene between

L = h(θ ^e )

^{and the desired labor supply}

N

^om

(w/p ^e )

^has

to be interpreted as a measure of a voluntary deviation from the ompetitive labor supply

of the workers, whih is a supply side measure. Similarly, any dierene between

L

^{and the}

desired ompetitiveemployment

h

^om

(w/p)

^{by the produer would be a demand side measure}

of voluntary deviationrelative tothe ompetitive regime.

Here, the voluntary underemploymentrate willbe dened in the usual way as

U = U

L, w p ^e

:= N

^om

(w/p ^e ) − L

N

^om

(w/p ^e ) = 1 − L

N

^om

(w/p ^e ) ,

⁽¹⁴⁾

whih measures the gap between the amount of labor whih is atually traded (i.e. worked)

and whih would be supplied by the workers under ompetitive onditions at the given wage

level. Sine the rate of unemployment is dened for all expeted real wages and all levels of

labor,

U

^{dened in(14) an alsobenegative. This ours forexample if}

w/p ^e

^{is relativelylow}

or

L

^{isrelativelyhigh. Weinterpretnegativeratesof}underemploymentasoveremployment (or overtime).

2.5 Nonompetitive Wage Setting versus Wage Bargaining

It isoftenonjetured thatnonooperativestrategibehaviorormarketpowerbyproduersor

by unions ould be a reason why unemployment in labor markets exists. This setion briey

presents the orresponding model with suh one-sided deviant behavior on the wage setting

and its impliation on the level of pries, wages, and on the level of employment 12

at given

ommodity pries. The omparison between the ooperative and nonooperative temporary

equilibria indued for the maroeonomywill be presented in Setion4.

The Monopsonisti Firm and Union Monopoly

Given

(p ^e , p) ≫ 0

^{and the aggregate labor supply funtion}

N

^om

(w/p ^e )

^{of workers, the monop-}

sonisti rm hoses a wage rate whihmaximizes

pF

N

^om

w

p ^e

− wN

^om

w

p ^e

.

This implies the rst-order ondition for aninterior solution

F ^′

N

^om

w

p ^e

= w p

1 + 1

E N

^om

(w/p ^e )

> w p

.

Let

w ˜ = W

^mon

(p ^e , p) = pW

^mon

(p ^e /p, 1)

^{denotetheuniquesolution,andlettheinduedaggregate}

employment and aggregate supply begiven by

L ˜ = h

^mon

p ^e

p

:= N

^om

W

^mon

(p ^e /p, 1) p ^e /p

, AS

^mon

p ^e

p

:= F

h

^mon

p ^e

p

.

The rst-order ondition implies that for any

(p ^e , p)

^,

h

^mon

p ^e p

< h

^om

p ^e

p

and

AS

^mon

p ^e

p

< AS

^om

p ^e

p

.

12

seealsoBöhm(2010)

Therefore, as a onsequene, at any given

(p ^e , p) ≫ 0

^{, the wage is equal to the marginal}

reservation wage of workers whih is smaller than the marginalvalue produt of laborfor the

rm. Thus, the rm reeives a monopsonisti surplus equal to

pF ^′ ( ˜ L) − w ˜ L ˜

^{, see Figure 3(a).}

However, atthe same time, the wage islarger than the true reservation wage.

PSfragreplaements

0

0 L

w

L ^∗ w ^∗

L ˜

˜ w

p ^e S

om

(L)

pF ^′ (L)

pF ^′ (L) 1+E S

om

(L)

om

(a)surplusofthemonopsonistirm

PSfragreplaements

0

0 L

w

L ^∗ w ^∗

L ˜

˜ w

om

pF ^′ (L)

om

p ^e S

om

(L)

1+E F ′ (L) p ^e S

om

(L)

(b)surplusofthemonopolistiunion

Figure3: Wages, employment,and surplus inmonopolistisituations;

(p ^e , p)

^given

Sine the produer aepts the market behavior of the workers as being given by their supply

funtion (whih orresponds to their marginal reservation wage), it seems as if the rm ould

exert more power and higher prots in the omparable bargaining situation by lowering the

wage to the true reservation wage, whih isnot an optionfor the produer tobe hosen under

market onditions. In other words, the employmentwage deision diers from the eient

bargaining under the most powerful bargaining situation for any given prie level

p

^{, when}

λ = 0

^.

The situation where a powerful union ontrols the labor market and sets the wage and the

employmentlevelisthesymmetrioppositeasetothemonopsonistirmandanbetreatedin

asimilarfashion. Given

(p ^e , p) ≫ 0

^{andthelabordemandfuntionoftheproduer}

h

^om

(w/p) = (F ^′ ) ^{− 1} (w/p)

^{, the} monopolisti union hoses a wage rate

w

^{whihmaximizes}

wh

^om

w

p

− p ^e S

h

^om

w

p

h

^om

w

p

= wh

^om

w

p

− p ^e n w

1 − τ w

v

h

^om

(w/p) n w

.

This implies the rst-order ondition

w p ^e

1

E h

^om

(w/p) + 1

= 1

1 − τ w

v ^′

h

^om

(w/p) n w

= S

^om

h

^om

w

p

withthesolution

w ˜ = W

^union

(p ^e , p) = pW

^union

(p ^e /p, 1)

^{whihinduesalevelofemploymentand}

aggregate supply

L ˜ = h

^union

p ^e

p

:= h

^om

W

^mon

p ^e

p , 1

, AS

^union

p ^e

p

:= F

h

^union

p ^e

p

.

For every

(p ^e , p)

^{, this indues a wage equal to the marginal value produt whih is, however,}

larger than the ompetitive wage and larger than the marginal willingness to work of every

worker atthe assoiatedlevelof employment. Thus, the workers obtainanaggregate monopo-

listisurplusequalto

pF ^′ ( ˜ L) − p ^e S

^om

( ˜ L)

^{,see Figure3(b). Asinthe aseof the}monopsonisti rm, the union aepts the labordemand behaviorby the produeras being given. Therefore,

the wage being equal to the marginalreservation wage of the produer is higherthan the true

reservation wage, equaltoaverage osts. Thus, atthe given prie,the powerfuluniondoesnot

obtain aess to the full rent from the produer, whih it ould obtain under bargaining and

λ = 1

^.

Summarizing the main results of this setion, one nds that the employmentwage deision

under one-sided strategi behavior in the labor market implies that the powerful side of the

market olletsan extra rent by exploiting the weaker trader, as is tobe expeted. Moreover,

this induesan ineient employment alloation sine the marginal willingness to work never

equalsthemarginalwillingnesstohiresineonlyonesideofthemarketisaprietakerwhilethe

otherone is not. Thisimpliesa lowerlevelofemploymentthan inthe ompetitivesituationat

allgivenpries and prieexpetations, whihisinontrast tothe eientbargainingsolution.

However, the strategibehavior does not generate unemployment.

13

3 Temporary Equilibrium with Eient Wage Bargaining

Itisnowstraightforwardtolosethemodelinordertodeterminethepropertiesofatemporary

equilibrium under wage bargaining. The data at the beginning of an arbitrary period are

aggregate money balanes

M > 0

^{held by oldonsumers, expeted priesfor the future period}

p ^e > 0

^{, and the bargaining parameter}

0 ≤ λ ≤ 1

^{, plus the parameters of the government}

(g, τ w , τ π )

^{. Then,atemporary}equilibriumwitheientwage bargainingisdened byapair of pries and wages

(p, w) ≫ 0

^{suh that the prie}

p

^{lears the ommodity market} ompetitively while the wage

w

^{equals the one set by the union and the produer inthe bargainingsolution.}

Assoiatedwiththe equilibriumistheequilibriumalloationwhihonsistsofapair offeasible

employment and output levels

(L, y ) = (L, F (L)) ≫ 0

^.

Sine allagentsin theeonomy onsumers,the produer,and the governmentare assumed

to be prie takers in the ommodity market, nding a temporary equilibriumis equivalent to

nding a prie

p

^{whih equalizes aggregate demand and aggregate supply, where aggregate}

demand hastobeappropriatelyadjustedtothe inomedistributioninduedbythe bargaining

result.

3.1 The Role of Union Power in Temporary Equilibrium

Aggregate Supply and Aggregate Demand

The bargaining wage

W (p ^e , λ, p, L)

^{and the assoiated employment level}

L = h(p ^e /p)

^were

derived asafuntionofprieexpetations andpriesintheprevioussetionwheretheemploy-

mentdeisionturned out tobeindependent ofthe bargaining parameter

λ

^{. Therefore, given a}

13

foramoredetaileddisussionseeSetion 4

pair of prie expetations and pries

(p ^e , p) ≫ 0

^{, the aggregate ommodity supply funtion is}

dened by

AS : R ++ → R ++ , AS(θ ^e ) := F (h(θ ^e )).

This is a funtion of the expeted ination rate alone, whih is globally invertible and dier-

entiable. Sine

h ^′ (θ ^e ) < 0

^{, one nds that}

AS ^′ (θ ^e ) < 0

^{so that, for any given prie expetation}

p ^e > 0

^{, aggregate supply isa stritly inreasing funtion of temporary ommodity pries}

d AS(p ^e /p) d p > 0.

In ontrast, the bargaining wage

W (p ^e , λ, p, h(p ^e /p))

^{will have an inuene on the inome}

distribution and thus on aggregate demand. Sine there are four dierent private onsumers

plusthegovernmentgeneratingaggregatedemand,theinomedistributionbetweenprotsand

wage inomeand the total inome generated determine aggregate demand.

The assumptionsonerningthe overlapping-generations struture ofonsumers implythatall

urrent net wages are saved and a proportion

0 ≤ c(θ ^e ) ≤ 1

^{of urrent net prots is onsumed}

by youngshareholders. Therefore, aggregate realdemand inanyperiodisthe sum oftotal real

moneybalanes

m := M/p

^{,governmentdemand}

g

^{,plusthe demandby} shareholderswhihisa funtion of aggregate prots. Thus, given money balanes, prie expetations, the bargaining

weight,andpries

(M, p ^e , λ, p)

^{,theinomeonsistentaggregatedemand}

y

^{mustbethesolution}

of

y = m + g + c(θ ^e )(1 − τ π ) π p

(13)

= m + g + c(θ ^e )(1 − τ _π )(1 − λ)

1 − E _F (L) E S (L) + 1

y

with

y = F (L)

^and

L = h(θ ^e )

^{. Therefore, one obtains as the} inome-onsistent aggregate demand funtion

y = D(m, θ ^e , λ) = m + g

1 − c(θ ^e )(1 − τ _π )(1 − λ)(1 − _E ^{E F (L)}

S (L)+1 )

= m + g

1 − c(θ ^e )(1 − τ _π )(1 − λ)(1 − _E ^{E F (h(θ e ))}

S (h(θ ^e ))+1 ) ,

(15)

whihisof the usualmultiplierformwith respet tomoneybalanes andgovernment demand.

Observe that aggregate demand is homogeneous of degree zero in

(M, p ^e , p)

^{. Therefore, for}

given

λ

^{,itisafuntionofrealmoney balanesandof theexpetedrate ofination. Obviously,}

∂D/∂m > 0

^{, i.e. real balanes have a positive eet on demand, and}

∂D/∂λ < 0

^{, i.e. higher}

bargainingpowerbytheuniondereasesprotsandthusonsumptiondemandbyshareholders.

In addition, if

∂D/∂θ ^e ≥ 0

^{, then the demand is stritly dereasing in the ommodity prie}

p

^,

i.e.

d D(M/p, θ ^e , λ)/d p < 0

^{is negative. This property holds in partiular when the savings}

proportionbyshareholdersisnondereasing andwhenthereservation wageandtheprodution

funtion are isoelasti.

Therefore,givenabargainingweight

0 ≤ λ ≤ 1

^andanypair

(M, p ^e ) ≫ 0

^{ofmoneybalanesand}

prieexpetations, thetemporaryequilibriumisgiven by aprie

p

^{whihlearsthe ommodity}

market, i.e.

D M

p , p ^e p , λ

= AS p ^e

p

.

⁽¹⁶⁾

Conerning existeneand uniqueness, one has the following immediate result.

Lemma 3.1 Let the aggregate supply funtion

AS

^{be globally invertible with}

AS ^′ (θ ^e ) < 0

^{, and}

assume that

∂D/∂θ ^e ≥ 0

^,

∂D/∂m > 0

^{hold. Then, forevery}

(M, p ^e ) ≫ 0

^and

0 ≤ λ ≤ 1

^{, there}

exists a unique positive temporary equilibrium prie

p > 0

^{solving equation (16).}

The uniqueness follows from the fat that the exess demand funtion is stritly monotoni-

ally dereasing. Figure 4 portrays the equilibrium situation in the usual aggregate demand

aggregate supply diagram of the ommodity market. As a onsequene of Lemma 3.1, one

PSfragreplaements

0

0 p

y

AS _p e

p

D

M p , ^p _p ^e , λ

Figure4: The temporaryequilibriumprie

obtains the following proposition.

Proposition 3.1 There exist dierentiable mappings

P : R ² ++ × [0, 1] → R ++

^and

W : R ² ++ × [0, 1] → R ++

^{, alled the prie law and the wage law} respetively, suh that

•

^{the unique positive temporary} equilibrium prie is givenby

p = P (M, p ^e , λ),

⁽¹⁷⁾

•

^{the unique positive temporary} equilibrium wage is dened by

w = W(M, p ^e , λ) := W

p ^e , λ, P(M, p ^e , λ), h

p ^e P (M, p ^e , λ)

,

and

• P

^and

W

^are homogeneous of degree one in

(M, p ^e )

^{, for given}

λ

^.

Properties of the Prie Law

Applying the impliitfuntion theorem to(16) with respet to

M

^{, one obtainsthe eet of an}

inrease of money balanes

∂ P

∂M =

1 P

∂D

∂m

− _P ^{p e} 2 F ^′ h ^′ + _P ^M 2 ∂D

∂m + _P ^{p e} 2 ∂D

∂θ ^e

> 0

PSfragreplaements

0

0 p

y

(a)inreaseofmoneybalanes

PSfragreplaements

0

0 p

y

(b)inreaseof

p ^e

^(for

∂D/∂θ ^e = 0)

Figure 5: Comparative-statiseets of money balanes and prie expetations

with anelastiity

0 < E P (M ) = ∂P

∂M M

P =

M P

∂D

∂m

− ^p _P ^e F ^′ h ^′ + ^M _P ^∂D _∂m + ^p _P ^e _∂θ ^∂D e

< 1.

⁽¹⁸⁾

Thus, the temporary equilibriumprie is a stritly inreasing and stritly onave funtion of

money balanes sine pries are nonnegative. Applying the impliit funtion theorem to (16)

one more, one obtainsa positiveexpetations eet onpries

∂P

∂p ^e = −

1 P F ^′ h ^′

− _P ^{p e} 2 F ^′ h ^′ + _P ^M 2 ∂D

∂m + _P ^{p e} 2 ∂D

∂θ ^e

> 0

with anelastiity

E P (p ^e ) = ∂P

∂p ^e p ^e

P = − _P ^{p e} 2 F ^′ h ^′

− _P ^{p e} 2 F ^′ h ^′ + _P ^M 2 ∂D

∂m + _P ^{p e} 2 ∂D

∂θ ^e

< 1,

⁽¹⁹⁾

whih is also less than one, implying that equilibrium pries are a stritly inreasing and

stritly onave funtion in prie expetations. Together this implies that the prie law

P

^is

stritly onave and homogeneous of degree one in

(M, p ^e )

^{, with a} representation of the form

p = p ^e P (M/p ^e , 1, λ)

^{whihis stritly inreasing and stritly onave in}

M/p ^e

^.

Output and Employment

Given the prie law, one obtainsthe assoiated temporary equilibriumalloation onsisting of

the levelsof output and employment asfuntions of the same data

(M, p ^e , λ)

^{, i.e.}

y = Y(M, p ^e , λ) := F

h

p ^e P (M, p ^e , λ)

and

L = L(M, p ^e , λ) := h

p ^e P (M, p ^e , λ)

.

(20)

whihare homogeneousof degree zero in

(M, p ^e )

^{. Using (18) and}

0 < E F (L) < 1

^{, one obtains}

the orresponding elastiitiesof money balanes on employment and outputas

E L (M) = −E h (θ ^e )E P (M ) > 0

^and

E L (M ) > E F (L)E L (M ) = E Y (M ) > 0.

⁽²¹⁾

Thus, higher money balanes implyhigher equilibriumpries but alsohigher levels ofemploy-

mentand output.

Similarly,applying property (19),

0 < E _F (L) < 1

^{,and the} relationship

E L (p ^e ) = E h (θ ^e )

| {z }

<0

(1 − E P (p ^e ))

| {z }

∈ (0,1)

< 0

⁽²²⁾

yields

E L (p ^e ) < E F (L)E L (p ^e ) = E Y (p ^e ) < 0.

Thus, output and employment deline with higher prie expetations. Therefore, ombined

with the zero-homogeneity of the employment law and output law, this onrms the tradeo

between money balanes and expetations for a onstant level of output and employment.

Figure 5 displays the omparative statis results for hanges of prie expetations and of real

money balanes.

Properties of the Wage Law

Inontrasttotheaboveresults,theomparativestatiseetsofthewagelawannotbesigned

in general sine several diverse eets interat in a nonlinear way. This an be seen partially

from the formof the wage lawequation

w = W(M, p ^e , λ) = λW Π P (M, p ^e , λ), L(M, p ^e , λ)

+ (1 − λ)W Ω p ^e , L(M, p ^e , λ)

,

⁽²³⁾

whih shows an interation of the eets of the prie law and the employment law in the

denition. However, it is possible in some speial situations to determine the eets under

more restrited onditions. Writing the wage as the assoiated mark-up over the reservation

wageoftheworkers(orequivalentlyasamark-downfromthereservationwageoftheproduer)

w =

1 + λ E S (L(M, p ^e , λ)) + 1 − E F (L(M, p ^e , λ)) E F (L(M, p ^e , λ))

W Ω (p ^e , L(M, p ^e , λ))

⁽²⁴⁾

=

λ + (1 − λ) E _F (L(M, p ^e , λ)) E S (L(M, p ^e , λ)) + 1

W Π (P (M, p ^e , λ), L(M, p ^e , λ)),

oneobservesthatthestatevariablesexert theirinueneonwages viaaprimaryeetthrough

theprieand employmentlaws andaseondaryeet throughthe respetiveelastiities, whih

determine the mark-up. Therefore, in situations where the eet of the state variable on the

mark-up is smalland an be negleted, the wage eet has the same sign as the employment

eet, i.e.

sgn E W (M ) = sgn E S (L)E L (M ) > 0

sgn E W (p ^e ) = sgn (E P (p ^e ) − (1 − E F (L))E L (p ^e )) > 0

(25)