Comparing Wages, Pries, and Payos - Efficient wage bargaining in a dynamic macroeconomic model

PSfragreplaements

0 0 L

w

F ⁻¹ (g) L

^rit

p = 0 w

^rit

p → ∞

Ω = 0 Π = 0

Figure13: Employmentwage pairs underindividual rationality and feasibility

of the prie funtion is zero and hanges sign, and where the prie and prot beome

in-nite. Thus, the set of bargaining pairs

(L, w)

^with ^positive ^prot ^onsists ^of ^the ^union ^of ^two

disjoint open regions allowing unbounded wages for

L < L

^rit ^and ^unbounded ^employment

levels.

As aonsequene one nds that the set of individuallyrational and

inome/demand-onsistent employmentwage pairs takes the form of a union of two adjoining sets as depited

in Figure 13. Observe that the two ritial employment levels, whih are the same for eah

stateof the eonomy

(M, p ^e )

^,^are ^determined^by ^demand ^features^and ^the^prodution^funtion.

They are independent of money balanes. However, high prie expetations may make the

lower ompat urvilinear triangle empty, implying that all equilibrium alloations must be

in the upper region of feasibility. Sine unbounded wages with unbounded pries are feasible

inome/demand-onsistentequilibriumalloationsforemploymentlevelsneartheupperritial

level, the assoiated set of payos must beunbounded and be equalto allof

R ² +

By adding the equilibrium points and the

λ

^-eieny ^frontier ^to ^the ^above ^diagrams, ^one

obtains inFigure 14a omparison of all senarios in alloationspae and inpayo spae. For

the isoelastiexample,allequilibriaareinthe ompattriangular regionofthe employment

wage spae. This shows also that the two one-sided strategi monopolisti situations indue

ineientemploymentlevelsbelowthe eieny frontier (leftpanelof Figure14). In ontrast,

theomparisoninpayospaeonrmstheloationofthetwoone-sidedmonopolistiequilibria

above the

λ

-bargainingfrontier, see Figure14(b). In otherwords, both monopolistiequilibria induebetter payoswhihannotbereahedorsupportedby theooperativedeisionsunder

eient bargaining. Notie, however, that the union's payo for

λ = 1

^is ^less ^than ^at ^the

nonooperativeequilibriumwhiletheproduer'sprotishigheratthenonooperativesituation

thanunderbargainingwith

λ = 0

^. ^However, ^these^relative^positions^of^the^payos^depend^on^the

prieexpetations. AsFigure15shows, thepayosinbothnonooperativeequilibriaarehigher

thanthemaximalpayosunderbargainingwhenexpetedpriesarehighenough. Theloation

in payo spae issurprising and ounterintuitive at rst. The arguments disussed at the end

of the two monopolisti ases show that, for eah given prie level

p

ⁱⁿ ^the nonooperative 15

Stritly speaking, the set also ontains the boundary point

(L

^rit

, w

^rit

)

^sine ^there ^exists ^an ^unbounded

intervalofpositiveprieswhihindue positiveprots.

PSfragreplaements

0 0

λ = 0 λ = 1

L w

F ⁻¹ (g) L

^rit

p = 0 w

^rit

p → ∞

Ω = 0 Π = 0

union

mon

(a)employmentwagepairs

PSfragreplaements

0 0

λ = 0 λ = 1

rit

union

mon

Ω

Π

union

mon

(b)payos

Figure14: Wages, employment, and payos under low prie expetations

situation,the monopolistan exert market power to obtain the full rent fromthe ompetitive

agent, a possibility whih neither the union nor the produer an obtain under bargaining.

Thus, the prie feedbak seems to wash out this eet under ooperation.

PSfragreplaements

0 0

union

mon

λ = 0 λ = 1

L w

F ⁻¹ (g) L

^rit

p = 0

w

^rit

p → ∞

Ω = 0 Π = 0

(a)employmentwagepairs

PSfragreplaements

0 0

union

mon

λ = 0 λ = 1

rit

Ω

Π

(b)payos

Figure15: Wages, employment, and payos under highprie expetations

The diagrams are drawn for the parameters of Table 1 and given values of the government

parametersandforgivenvaluesofthestatevariablesmoneybalanesandexpetations. Beause

of ontinuity, these features are loally robust properties and they will be observed for this

isoelasti lass of models in dierent magnitudes and possibly also in dierent relative orders

under dierent parameters and values of the state variables. However, as some numerial

experiments have shown, the basi features are preserved for a wide range of values of the

parameters and of state variables. The overall homogeneity of the prie law and the wage law

does not prelude reversals oropposite eets.

While these result might seem to be ounterintuitive at rst sight, it is straightforward to

disern the two prinipal reasons why these eets our. First of all, the maximization of

nominalobjetives(protresp.exess wages)reatesspilloversbetween markets even forstati

general-equilibriumsystems,whihareprimarilyduetoinomeeets. Beauseoftheseinome

eets, it isunlikelythat the universal omparative-statisresults (asoftenderived in

partial-equilibrium models with strategi behavior) will persist in general-equilibrium models. It is

knownfromgeneralequilibriumtheorythatsuheetsareduetoprienormalization,implying

dierent real alloations, relative pries, and nominal values of inomes (prots and wages)

underdierenthoiesofanumeraireorofprieindexes. Theseresultsarewelldoumentedand

havebeenreognizedinmanydierentontextsinpartiularinwelfareeonomis,international

trade, or oligopoly theory whenever inome feedbaks are taken into aount appropriately

with a nononstant marginalutility of inome for onsumers.

In temporary equilibriumof a

monetary eonomy, these eets learly do not disappear.

Seond, the prie feedbak, whih was shown to be responsible for the ineieny of the

bar-gaining solution under ompetitive prie taking in temporary monetary equilibrium, operates

in eah of the three ases endogenously in a dierent way. There is no strutural feature of

the modelwhih relatesthe nominal payos, hosenfor the bargainingproblem neitherto the

nominal objetives by the monopolist/monopsonist with wage setting and prie taking nor to

the results indued by the maximizationunder ompetitive prie and wage taking. Thus, in

all three ases, the prie feedbak and the inome feedbaks have a deisive inuene on the

nominal values hosen for the payos in the monetary eonomy. For these reasons, the four

labormarketsenarioswhose equilibriumharateristis areompared inthepriewage spae

and in payo spae are in general not omparable with respet to real alloationsor nominal

payos, even under the weak onept of eieny. Sine, in addition, equilibrium pries and

alloations depend on the other state variables, anextensive welfare analysis may not lead to

onlusive results.

It is worth noting that some properties of the results are spei to the isoelasti model

ho-sen for the numerial analysis sine the bargaining parameter

λ

^plays ^a ^spei ^dual ^role ⁱⁿ

temporary equilibrium. On the one hand, there is no impat of union power on aggregate

supply. Therefore, the interation of the isoelasti struture between prodution and labor

supply shows that the measure of union power

λ

^exerts ^a ^diret ^inuene ^on ^the ^real ^wage

mark-up and on the level of underemployment, making both of them onstant in temporary

equilibrium. Theseonstantsdepend ontheelastiitiesof thelabormarketpartiipantsandon

union poweronly. Thus, in a dynami eonomy as analyzed inthe next setion, both of them

are onstant over time, i.e.independent of

(M, p ^e )

^, ^and ^they ^are independent of allsal and demand parameters in the eonomy. On the other hand, a powerful union whih an hoose

the parameter

λ

^does^not^exert ^absolute^ontrol^over^its^seemingly^most ^important^endogenous

variable the wage rate. Moreover, even for the isoelasti ase, it seems unlear whether the

wage outome under bargainingdominates the ompetitiveoutome, insome other sense than

theeieny riterionusedabove. Itremainsanopenquestiontowhatextenttheineienies

will hange or disappear if the bargaining agents hose real rather than nominal payos as

objetives.

seeforexampleDierker&Grodal(1986);Böhm(1994);Gaube(1997);Roberts&Sonnenshein(1976)

5 Dynamis of Monetary Equilibrium

So far the harateristis of equilibria under bargaining were disussed for an arbitrary given

period

t

^with ^initial ^money ^balanes

M t

^held ^by ^the ^private ^setor, ^expeted ^pries ^for ^the

next period by onsumers

p ^e _t,t+1

^, ^and ^by ^the ^union ^power

λ t

^. ^Thus, ^the ^triple

(M t , p ^e _t,t+1 , λ t )

desribesthe stateof theeonomy atany given time. Assoiatedwitheahstate arethe pries

and wages and the levels of output and employment

(p t , w t , y t , L t )

ⁱⁿ ^temporary equilibrium whihare dened by applying the respetive mappings fromthe previoussetion.

Thissetionanalyzesthedynamibehavioroftheeonomy inequilibriumassumingthatunion

power is onstant over time and given exogenously at some level

0 ≤ λ ≤ 1

^. ^Sine

λ/(1 − λ)

determines the relative share of wages over prots, no other eonomi variables related to

the objetives of the agents are onsidered. As was shown in the previous setion,

λ

^has ^a

signiantimpatonmost importanteonomi variablesineveryperiod,likeoutput,inomes,

pries, and onsumption, whih are relevant for welfare. Thus, it would be desirable to relate

thespeivaluehosenforunionpowertothemarketdatawhihareinduedandtoreevaluate

the equilibrium outome with respet to the true objetives of the agents. This leads to an

endogenous determinationof the measure of bargainingpower. Forthe dynamis, this implies

that an adaptive rule or a dynami mehanism has to be dened based on the data in eah

period. However, at this stage we examine the dynamis of the monetary eonomy without

providing any justiation what level of union power

λ

^would ^be ^reasonable ^to ^be ^assumed,

leavingsuhquestionstobeaddressedinfuture researh. Therefore, the dynamidevelopment

of the eonomy will be desribed ompletely by haraterizing the evolution of the two state

variables money balanes and expeted pries

(M t , p ^e _t,t+1 )

^, ^implying ^a two-dimensional state spae

X := R ² ++

5.1 Perfet Foresight

A sequene

{p ^e _t,t+1 , p t } ^∞ _t=t ₀

^of ^pries ^and expetations will be said to have the perfet-foresight propertyif

p ^e _t,t+1 = p _t+1

^holds^for^all

t

^. ^It^is^one^of^the^main^questions^of^dynami^maroeonomi

analysistondonditionsanddenetheoneptswhihensurethatperfet-foresightsequenes

are in fat generated by an assoiated dynamial system whih is globally dened. In other

words, a foreasting rule ora preditor has to be dened to ensure perfet foresight alongany

orbit.

In order to guarantee that, for any period

t

^, ^the ^atual ^prie

p t

^oinides ^with ^its

assoiated predition

p ^e _t − 1,t

^, ^the ^ondition

p ^e _t−1,t = P (M t , p ^e _t,t+1 , λ)

must hold for any

t

^. ^This ^denes ^impliitly ^the ^funtional relationship determining how the foreast inany periodfor the next one should behosen as afuntion ofthe previous foreast.

Therefore, solving (16) for the expeted prie

p ^e _t,t+1 = ψ ^∗ (M t , p ^e _t − 1,t , λ) ≡ P ^e M t , p ^e _t − 1,t , λ

:= p ^e _t − 1,t AS ⁻¹

D M t

p ^e _t−1,t , λ

We will assume throughout this setion that the aggregatedemand funtion is independent of expeted

ination. ThegeneralaseouldbedealtwitheasilyusingtheresultofLemma3.1.

seeBöhm(2010)

denes the perfet preditor

ψ ^∗ (M t , ·, λ)

^sine ^for^all

(M t , p ^e , λ) P(M t , P ^e (M t , p ^e , λ), λ) = id (M t ,λ) (p ^e ).

Therefore, the two mappings

M t+1 =M(M t , p ^e _t − 1,t , λ) := M t + p t (g − τ D ˜ (M t /p t , λ)) p ^e _t,t+1 =ψ ^∗ (M t , p ^e _t−1,t , λ)

(50)

with

p t = P (M t , ψ ^∗ (M t , p ^e _t−1,t , λ), λ)

^and

˜ τ ≡ τ ˜

p ^e _t,t+1 p t

, λ

= ˜ τ

ψ ^∗ (M t , p ^e _t − 1,t , λ)

P (M t , ψ ^∗ (M t , p ^e _t − 1,t , λ), λ) , λ

denethedynamibehaviorofmoneybalanesandexpetationsunderperfetforesightforany

level of bargaining power

λ

^. ^In ^addition,

τ ˜

^denotes ^the ^average ^tax ^rate ^whih ^will^be ^derived

in(52). Sine for all

t

^, ^one ^has

p ^e _t − 1,t = p t

^, ^one ^an ^rewrite ⁽⁵⁰⁾ ^as

M t+1 = M(M t , p t , λ) =M t + p t (g − τ D ˜ (M t /p t , λ)) p _t+1 = ψ ^∗ (M _t , p _t , λ) =p _t AS ⁻¹

D

M t

p t

, λ

,

(51)

dening equivalent dynamis with perfet foresight in the spae of money balanes and pries

(M, p)

^for ^any ^given ^level

λ

^of ^bargaining ^power.

It is one of the reurring themesof dynamialeonomies with prie expetations that in most

ases prie dynamis indued under perfet foresight are unstable, a phenomenon whih also

oursintheurrentmodel. Toseethis, let

M > ¯ 0

^denote^an^arbitrary^onstant^level^of^money

balanesand

λ

^be^given. ^Then,⁽⁵¹⁾^redues ^to^the one-dimensionaldynamialsysteminpries

G : R ++ → R ++

p t+1 = ψ ^∗ M , p ¯ t , λ

=: G (p t ) .

Rewriting (16) one nds that ithas the unique positive xed point

p = M ¯

D ⁻ ¹ (AS(θ ^e ); λ) = M ¯

D ⁻ ¹ (AS(1); λ) ,

where

D ⁻¹

^is ^the ^inverse ^of ^the ^aggregate ^demand ^funtion ^with ^respet ^to ^its ^rst ^argument

M/p

^. ^Sine ^the ^prie ^law ^is ^invertible ^with ^respet ^to ^prie expetations with an elastiity stritlybetween

0

^and

1

^implies^that^the^unique^positive^xed^point

p

^isasymptotiallyunstable sine

G ^′ (p) = ∂ψ ^∗

∂p ^e ( ¯ M, p, λ) = ∂P ^e

∂p M , p, λ ¯

= 1

∂ P

∂p ^e ( ¯ M , ψ ^∗ ( ¯ M , p, λ), λ) > ψ ^∗ ( ¯ M, p, λ)

P ( ¯ M , ψ ^∗ ( ¯ M , p, λ), λ) = 1.

Im Dokument Efficient wage bargaining in a dynamic macroeconomic model (Seite 38-44)

Comparing Wages, Pries, and Payos

0

0 L

w

F −1 (g) L

p = 0 w

p → ∞

Ω = 0 Π = 0

(L, w)

L < L

(M, p e )

R 2 +

λ

λ

λ = 1

λ = 0

p

(L

, w

)

0 0

λ = 0 λ = 1

L w

F −1 (g) L

p = 0 w

p → ∞

Ω = 0 Π = 0

0 0

λ = 0 λ = 1

Ω

Π

0 0

λ = 0 λ = 1

L w

F −1 (g) L

p = 0

w

p → ∞

Ω = 0 Π = 0

0 0

λ = 0 λ = 1

Ω

Π

λ

λ

(M, p e )

λ

t

M t

p e t,t+1

λ t

(M t , p e t,t+1 , λ t )

(p t , w t , y t , L t )

0 ≤ λ ≤ 1

λ/(1 − λ)

λ

λ

(M t , p e t,t+1 )

X := R 2 ++

{p e t,t+1 , p t } ∞ t=t 0

p e t,t+1 = p t+1

t

t

p t

p e t − 1,t

p e t−1,t = P (M t , p e t,t+1 , λ)

t

p e t,t+1 = ψ ∗ (M t , p e t − 1,t , λ) ≡ P e M t , p e t − 1,t , λ

:= p e t − 1,t AS −1

D M t

p e t−1,t , λ

ψ ∗ (M t , ·, λ)

(M t , p e , λ) P(M t , P e (M t , p e , λ), λ) = id (M t ,λ) (p e ).

M t+1 =M(M t , p e t − 1,t , λ) := M t + p t (g − τ D ˜ (M t /p t , λ)) p e t,t+1 =ψ ∗ (M t , p e t−1,t , λ)

p t = P (M t , ψ ∗ (M t , p e t−1,t , λ), λ)

˜ τ ≡ τ ˜

p e t,t+1 p t

, λ

= ˜ τ

ψ ∗ (M t , p e t − 1,t , λ)

F ⁻¹ (g) L

(M, p ^e )

R ² +

F ⁻¹ (g) L

F ⁻¹ (g) L

(M, p ^e )

p ^e _t,t+1

(M t , p ^e _t,t+1 , λ t )

(M t , p ^e _t,t+1 )

X := R ² ++

{p ^e _t,t+1 , p t } ^∞ _t=t ₀

p ^e _t,t+1 = p _t+1

p ^e _t − 1,t

p ^e _t−1,t = P (M t , p ^e _t,t+1 , λ)

p ^e _t,t+1 = ψ ^∗ (M t , p ^e _t − 1,t , λ) ≡ P ^e M t , p ^e _t − 1,t , λ

:= p ^e _t − 1,t AS ⁻¹

p ^e _t−1,t , λ

ψ ^∗ (M t , ·, λ)

(M t , p ^e , λ) P(M t , P ^e (M t , p ^e , λ), λ) = id (M t ,λ) (p ^e ).

M t+1 =M(M t , p ^e _t − 1,t , λ) := M t + p t (g − τ D ˜ (M t /p t , λ)) p ^e _t,t+1 =ψ ^∗ (M t , p ^e _t−1,t , λ)

p t = P (M t , ψ ^∗ (M t , p ^e _t−1,t , λ), λ)

p ^e _t,t+1 p t

ψ ^∗ (M t , p ^e _t − 1,t , λ)

P (M t , ψ ^∗ (M t , p ^e _t − 1,t , λ), λ) , λ

p ^e _t − 1,t = p t

M t+1 = M(M t , p t , λ) =M t + p t (g − τ D ˜ (M t /p t , λ)) p _t+1 = ψ ^∗ (M _t , p _t , λ) =p _t AS ⁻¹

p t+1 = ψ ^∗ M , p ¯ t , λ

D ⁻ ¹ (AS(θ ^e ); λ) = M ¯

D ⁻ ¹ (AS(1); λ) ,

D ⁻¹

G ^′ (p) = ∂ψ ^∗

∂p ^e ( ¯ M, p, λ) = ∂P ^e

∂p ^e ( ¯ M , ψ ^∗ ( ¯ M , p, λ), λ) > ψ ^∗ ( ¯ M, p, λ)

P ( ¯ M , ψ ^∗ ( ¯ M , p, λ), λ) = 1.