The Suntory and Toyota International Centres for Economics and Related Disciplines
Insider Power, Unemployment Dynamics and Multiple Inflation Equilibria
Author(s): Ben Lockwood and Apostolis Philippopoulos
Source: Economica, New Series, Vol. 61, No. 241 (Feb., 1994), pp. 59-77
Published by: Blackwell Publishing on behalf of The London School of Economics and Political Science
and The Suntory and Toyota International Centres for Economics and Related Disciplines
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Economica(1994) 61, 59-77
InsiderPower, UnemploymentDynamics
and Multiple InflationEquilibria
and APoSTOLISPHILIPPOPOULOS
By BEN LOCKWOOD
University of Exeter and University of Essex
Final versionreceived7 February1993.
This paperincorporatesemploymentdynamicsdue to insiderpowerinto the Barro-Gordon
theoryof inflation.Unlike in the static Barro-Gordoncase, and even though our infinitehorizonmodel is linear-quadratic,
therecan be two Markov-perfectequilibria,so that the
sameunemployment
maycorrespondto eitherlow or highinflation.Bothequilibriuminflation
rates (measuredper unit of the gap betweenthe insiders'and government'semployment
ratein the staticcase.The low inflationequilibrium
targets)arehigherthanthe corresponding
has intuitivecomparative-statics
properties,in that inflationis increasingin insiderpowerand
the discountfactor,but the high inflationequilibriumbehavesin the oppositeway. The highinflationequilibriumcan be eliminatedby a (future)precommitment
to a fixedinflationregime
(e.g. ERM), but only if this precommitmentcan be made with certainty.Contraryto the
staticmodel,inflationmay be lowerwhenthe unionputs a positiveweighton a wagetarget
than whenit cares only about employment,even when this wage targetis consistentwith a
loweremploymentlevel than the employmenttarget.
INTRODUCTION
In their well-knownpaper of 1977, Kydlandand Prescottbrieflydiscusseda
theoryof inflationthat was consistentwith rationalbehaviouron the part of
both wage-settersand government.This theory was subsequentlydeveloped
and extended'by Barroand Gordon (1983a,b), Canzoneri(1985), Horn and
Persson (1988) and Tabellini(1988), among others. In its originalform, the
theory relies on the assumptionthat employmentequilibriumis below the
government's employment target if, 'for example, . . . the distortionsfrom income
taxation, unemployment compensation and the like make the average level of
privately chosen work and production too low' (Barro and Gordon 1983b). A
moreplausibleexplanationof the divergence,at least in the Europeancontext,
is that wages are set by a tradeunion, and the employmenttargetof the trade
union consists only of the currentmembership,or at least puts lower weight
on those in the labourforcewho are not in the tradeunion. This interpretation
of the Barro-Gordon (BG) model has been suggested by, among others,
Canzoneri(1985), Horn and Persson(1988) and Tabellini(1988).
However,empiricalwork on trade union membership(e.g. Booth 1983)
suggests that membershipis not fixed over time, but responds strongly to
changesin past employment.Thereis a recenttheoreticaland empiricalliterature which attemptsto use these membership,or insider,dynamicsto explain
unemploymentpersistence(Blanchardand Summers1986; Kidd and Oswald
1987;Carruthand Oswald1987;Alogoskoufisand Manning1988;Lockwood
and Manning1989;Layardand Bean 1989;Nickell 1990and Blanchfloweret
al. 1990). However,to our knowledgeno one has formallyincorporatedthese
dynamicsinto the BG model.
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In this paper,we proposea simplegeneralizationof the BG model which
incorporatesthese dynamics.Specifically,following Blanchardand Summers
(1986), we suppose that the currentstock of union members,or insiders,is
equal to last period'semployed,and the employmenttargetof the union is a
weightedcombinationof insidersand the whole labourforce.This impliesthat
(for example)an adverseshock whichreducescurrentemploymentwill reduce
next period's union employmenttarget and thereforenext period's actual
employment.Therefore,there is persistencein (un)employmentover time; in
this setting, an unanticipatedinflationat time t will reducenot only current
unemployment,but alsofutureunemployment.A rationalgovernmentwilltake
into accountthese futureeffectsof currentpolicy.
So, we model the interactionbetweengovernmentand privatesector as a
dynamicgame,with laggedunemploymentas the statevariable.FollowingBG,
we focus on the "discretionary"equilibrium,whichis a credible,or subgameperfect,Nash equilibriumin the dynamicgame, where:(i) within any period,
the governmentcannot precommitto an inflationrate beforewages are set by
the union; (ii) "punishmentstrategies"are ruledout by restrictingthe agents'
actions to depend on the past only through the current value of the
state variable.Any subgame-perfectequilibriumsatisfying(ii) is known as a
"Markov-perfect"
equilibriumin the game-theoreticliterature(Fudenbergand
Tirole 1991).
Our resultsare of two kinds. The first concernsthe numberof equilibria.
Unlike in the staticBG model, and even thoughthe model is linear-quadratic,
therecan be multiple(two) equilibria:the same level of unemploymentcan be
associatedwith eitherhigh or low inflation.Furthermore,we developan intuitive explanationfor this; the reactionfunctionof the government,appropriately
defined, is nonlinear.The possibility of multiple Markov-perfectequilibria
in linear-quadraticinfinite-horizongames has been noted before,2first by
Papavassilopoulosand Olsder(1984), althoughthey did not presenta specific
example.Obstfeld(1991) presentssuch an examplein a publicfinancemodel;
however,his modelis rathercomplexwith severalstatevariables,andmultiplicity requiresthat government'sdiscountrate exceedsthe privatesector's.Our
model, by contrast,is extremelysimple.In addition,unlikemany macroeconomic rationalexpectationsmodelswithmultipleequilibria,we cannoteliminate
all but one of the equilibriaon groundsof instability.3Thatis, herethe employment and inflationdynamicsare stable in both equilibria.The questionthen
arises as to how-if at all-the bank can select the (Pareto-superior)lowinflationequilibrium.We show that, if the bankcan makea futureprecommitmentto a fixed-inflationregime(e.g. the ExchangeRate Mechanism,or ERM),
then the high-inflationequilibriumis eliminated.Furthermore,if the ERM rate
of inflationis lower than in the low-inflationequilibrium,then inflationwill
fall monotonicallyuntil ERM entry takes place. However, this device will
work only if the governmentcan precommitto the fixed-inflationregimewith
probability1; precommitmentwith probabilityq< 1 is not enough.
The secondset of resultsof the paperinvolvecomparisonsof the properties
of the dynamicmodel to the static BG model. First, we examinethe simple
case wherethe union has only an employmenttarget.There,in both equilibria,
the inflationarybias is always higher than in the BG model. The reason for
this is straightforward:
in the dynamiccase a smallincreasein currentinflation
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not only reducestoday'sunemploymentrate,but also reducesfutureratessince
unemploymentdisplayspersistence.So the incentiveto inflatetoday is higher.
Furthermore,the two equilibriapossessdifferentcomparative-statics
properties
to one another;the low (high)inflationrateis increasing(decreasing)in insider
power and the discountfactor. So, the behaviourof the low-inflationequilibriumis intuitivelyplausible,but the high-inflationequilibriumbehavescounterintuitively.4
We also consider a numberof extensionsof the basic model. The most
importantof these is to allow the union to have a real-wagetarget.We show
thatinflationcan be lower(conditionalon a givenunemploymentrateinherited
from the past) when the union puts a positive weight on a wage target than
whenit caresonly about employment,even when this wage targetis consistent
with a lower level of employmentthan the employmenttarget. The intuition
for this is as follows. The degreeof unemploymentpersistencefalls when the
union cares about real wages, so that any reductionin the currentrate will
have less effect on future rates. This reducesthe incentiveto inflatetoday in
orderto reducethe currentrate. This is the oppositekind of predictionto the
static BG model wherethe introductionof a wage targetincreasesthe (static)
unemploymentrate and hence inflation.Finally, we show that all the above
resultscan be generalizedto stochasticor continuous-timestructures.
A full discussionof relatedwork is given in the main body of the paper.
However,it is worth noting here that, althoughtherehas been some previous
work on monetary policy games with structuraldynamics (see Miller and
Salmon 1985; Oudiz and Sachs 1985; and Levine 1988), these paperstend to
study ratherdetailedmacromodels, and must resortto numericalanalysisto
get results.Moreover,they seemto have overlookedthe possibilityof multiple
equilibria.Our model is so simple that we can usually get analyticalresults;
and wherewe have to resortto numericalanalysisthe intuitionis alwaysclear.
Also, our papershowshow multipleequilibriamay arisein modelsof this type.
The restof the paperis organizedas follows.SectionI introducesthe model.
SectionII analysesthe discretionaryequilibriumfor the case wherethe union
has only an employmenttarget.SectionIII considersthe questionof how the
bank can select among multipleequilibria.SectionIV considersextensionsof
the model to allow for a real-wagetargetfor the union and stochasticshocks,
and also examinesthe continuous-timeversionof the model.
I.
THE MODEL
The model is a very simple one, which can be interpretedas a generalization
of Barroand Gordon (1983a). The economy is composedof a privateand a
public sector, which interact over an infinite number of time-periods,t=
1, 2, .... The privatesector consists of a trade union which can unilaterally
set the nominalwage, WQ,in any time period t. The public sectoris a central
bank whichchooses the pricelevel, P,, in everyperiodby manipulatingeither
the money supply or the exchangerate.5The level of employmentin period
t, L,, is then determined by the real wage, WI/Pt, as L = (WI/P,)-,
(1)
lt =
.(pt
wt)
,
where,in what follows, we set E= 1 for convenience.
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The trade union's payoffs are as follows. In the spirit of Lindbeckand
Snower (1986), and following Blanchardand Summers(1986) and Alogoskoufis and Manning (1988), we assume that the employmenttarget of the
union, Lu, is the weightedgeometricmean of those 'insiders'who have been
recently employed, Lt-I, and the total labour force, N. Thus, Lu= L 1N- a
where0 < a < 1. Then, takinglogarithms,we get:
(2)
t=1,. .., oo,
it=alt- I+(1-a)n,
whereintialemployment,10,is predetermined.
We also allow the union to have
a real-wagetarget, Q, and to dislike inflation, i.e. to have a zero inflation
target.Thus, the payoffto the union in the entiregame dependsnegativelyon
deviationsof actual employment,It, and real wages, wt -pt, from targets Iu
and Q, respectively, and on inflation.
00
(3)
I_XE
Z
t- l {(lu _4)2 + 0[r - (Wt_pt)]2+P(pt _pt_. )2},
t= I
where 0< 8 < 1 is the discount factor, and 0 and p are the relativeweights
givento realwagesand inflationrelativeto employment.We shallfirstanalyse
the case where the union cares only about employment,0 = p = 0. This case
keeps the expositionsimpleand also follows BG. More generalcases, with 0
and p $0, will be consideredbelow in SectionIV.
Next, considerthe centralbank. It has both an employmenttarget,1*,and
an inflationtargetof zero. Its payoffsin the entiregame are:
(4)
B=-2
00
E
(
(Pt
1)2]9
t=I
where0?< ?<1is the relativeweight that the bank places on the employment
target.We assumethat 2>0. The role of this assumptionis discussedbelow
in Section11(e).
Withinany time period,the orderof eventsis as follows. First, the insiders
choose a nominalwage level, wt, and followingthis the bankchoosesthe price
level,pt. Thisorderof eventscorrespondsto what BG call 'discretionary'
policy
on the part of the bank. Finally, given the resultingreal wage, the level of
employmentis given by (1).
The model just outlinedis very similarto Barro and Gordon (1983a,b);
in fact, in the specialcase of a = 0, so that last period'semployeddo not affect
the insider'starget at all, and the assumptionthat l* > n, we obtain precisely
the BG model. However,for most of the paperwe set P*= n so that the bank's
most desiredlevel of employmentis full employment.6It then follows from (2)
that asymptotically,as t-+oo, the insiders'and the bank'semploymenttargets
coincide. Therefore,when 0= p = 0, our model does not exactly generalize
the BG model; we have a temporary,ratherthan permanent,divergenceof
employmenttargets.This simpliesthe algebra,and it is still possibleto make
meaningfulcomparisonswith BG.7
Also, the incorporationof laggedemploymentinto the union'semployment
target(2) above makesquite an importanttechnicaldifference;when a > 0, it
convertsthe game from a repeatedgame into a dynamicgame with a state
variable, It. Our approach in the rest of this paper is to explore how the
equilibriuminflationrate is changedby these employmentdynamics.
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II.
DISCRETIONARY EQUILIBRIUM
In BG's repeated-gamemodel, the discretionaryequilibriumis simplythe oneshot Nash equilibriumin the constituentgame where,within the period, the
bank cannot precommitto the rate of inflation before the insiders set the
nominalwage. In this more generalsettingit is the perfectequilibrium,where
the currentactions of the players at time t, namely (pt, we), depend on the
game history only throughthe state variable,I,_-, often known as Markovperfectequilibrium(Fudenbergand Tirole 1991)in the gametheoryliterature.
In what follows,we adopt the terminology'Markov-perfect'
for the discretionary equilibrium.
(a) Characterizationof Markov-perfect equilibrium
Note firstthat, in the absenceof stochasticshocks, the insiderscan alwaysset
the wage in any period t to ensurethat they achievetheiremploymenttarget
lt, conditionalon their expectationof the price levelpt. Thus, they set w, so
where 7rt=pt-pt-P and
that lu=p'-w,.
Then, from (1), lt-lu=crt-7r',
It is convenient to suppose that the action variable of the union
? =e=P-Pe ptin each period is its inflationexpectation, re, and the action variableof the
bank is ,rt. (Of course, in equilibrium,,re= irt.) We now transformthe state
variableto unemployment,ratherthanemployment.Definethe unemployment
rate at time t to be ut= n -
lt,'
i.e. the difference between the logs of full and
actualemploymentat t. The dynamicsof ut are given by the state equation:
(5)
t=1,.
-
ut=n-lt=(n-Il)+(lu-lt)=autI-(rt
. .,
00,
Ie),
where the initial unemployment rate, uo= n - 10, is predetermined. Note that
(5) tells us that thereis persistencein unemploymentif a > 0.
As the model is linear-quadraticand we are looking for an equilibrium
wherecurrentactions dependonly on the state variableut_I, we can assume
that the players'actionswill be linearfunctionsof uti-; i.e.,
(6)
?t= ZtUt-X'
Ze=
reu
where r, ire are coefficientsto be determined.
A linear Markov perfect equilibrium(LMPE) is then a pair (7r,,re) that are
mutualbest responsesfor every possible ut-I. This can be definedmore formally. Let V'(ut -), i= b, u be the discounted present value of losses to the
bankand union respectivelyfrom t onwards,givenan unemploymentrateut- I
inheritedfrom t -1.9 Then for a LMPE we requirethat: (i) ir maximizes
Vb(ut_), given (5), (6) and ,refixed; (ii)
,re
maximizes Vu(ut-1), given (5), (6)
and ir fixed. Now from the linear-quadraticstructureof the model, the
functionsV' will be quadraticin u,_- ; i.e., Vi(u,_ 1) = -fiu2_ 1/2, i= b, u, where
the ,' are as yet unknownparameters.However,we can note at this point that
the union has only one target, ir, and one instrument,ire; therefore,it can
simply set ,re= ir in every period (rationalexpectations),thus achievingzero
loss in every period. This suggests that in equilibrium f3U = O. So from now on
we set flu= 0, and confirm that this is indeed the case in equilibrium. Then, we
also set pb
=
P6.Finally,as the bank'slosses are non-positive,P>0.
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If ,r, ,ceare mutual best responses'0 given (5), then (,B , T) must satisfy
the following equations:
(7)
- min (A + Sf3) - + (1 - A) -
2
(8)
ir2
Ar
2
given (5), (6), and ire fixed
AT
The left-hand side of (7) is the present value of losses from t onwards, and the
right-hand side is the current value of losses, plus the losses from t + 1 onwards,
evaluated at the LMPE levels of actual and expected inflation.
We solve (7) and (8) for the LMPE (,r, ,T) in two stages. First, we find the
reaction functions of the two players and consequently the Nash equilibrium
level of inflation, conditional on afixed B; and then we solve for ,B.The reaction
functions, conditional on fixed PB,are then as follows. First, the' first- and
second-order conditions in (7) are -(A+ 813) [a - ( -ire)] + (1 - A)r = 0,
1 + 6p > 0 respectively. As 8, B2 0, the second-order condition for a maximum
always holds. From the first-ordercondition,"Iwe obtain the following reaction
function for the bank, conditional on P:
(9)
,r(r e)
=(A+ (6p)(a +
1+ Sf3
)
Finally, it is obvious from (8) that the union's reaction function is
(10)
Te(ir) =r.
These reaction functions always have a unique intersection at
(11)
,Tr(p)=,re(1)
a,
which is the Markov equilibrium level of inflation per unit of unemployment,
conditional on ,B.
The coefficient P is now solved as follows. Substitution of (11) into (7)
yields the Riccati equation defining ,B:
a282p2
+ [- _ 1 + (1 + A)35a2]
+
a2 = 0.
Multiplying the Riccati equation through by S and re-parameterizing, we get:
(12)
pi2+[)-1+(1+A)p]7j+)p=O,
ij=8fl,p=Sa2.
The solution(s) il to the Riccati equation will give us equilibrium inflation via
the formula in (11). Appendix A shows that (12) has at least one non-negative
real solution if and only if the following holds:
(13)
+A)-2A<P2(1
so that (13) is necessary and sufficient for existence of a LMPE. It holds if p=
8a2 is low enough.
(b) Comparisonwith the static BG model
From (11), inflation at time t is simply irt=irut-I=[(X+
p)/(I-A)]aut-i.
From (5), ut= aut -I, so inflation per unit of the current unemployment rate,
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ut, is 7C= [Q+ P)/(1 -AX)]. In the BG model, by contrast, the equilibrium
unemploymentrateis fixedat someuoin everytimeperiod.In thiscase,inflation
per unit of the unemploymentrate is 7rbg=- /(1 - A). Comparing7Crand ir ,
we see that, as long as / > 0, the inflationarybias perunit of the unemployment
in
rate is higherin the dynamicmodel. The reasonfor this is straightforward;
the dynamicmodel,a smallincreasein currentinflationnot only reducestoday's
unemploymentrate,but in equilibriumit also reducesfutureratessinceemploymentgainspersist.So thereis an additionalincentiveto go for higherinflation.
(c) Multiple equilibria
Assumethat (13) holds, and let the smallerand the largerof the solutionsto
(12) be il and i] respectively.As long as p < p, l < i]. Also, since ji = )p > 0,
as long as p satisfies(13) with a strict inequality,there are two positive real
solutions to (12). Each correspondsto a LMPE of the model; so we can
concludethat, as long as p < p, we have two equilibriuminflationlevels,127f
and #s, with r < #f. Uniquenessarises, in general,only if the union and the
government have the same employment target (n =
ln),
which requires a = 0.
In this case, the model is reducedto a naturalrate one with a zero inflation
bias.
One possible explanationfor multiple equilibriais as follows. It can be
shown that the unconditionalreactionfunction of the bank (i.e. the reaction
function (9) with ,B eliminated)is nonlinear,intersectingthe (unconditional)
)r)rI
r)
e
e 1)
//~~~~~
450
FIGURE 1
reactionfunctionof the union twice, as shown in Figure 1. So the multiplicity
arises from nonlinearitiesin the reactionfunctions-appropriatelydefinedas in the strategiccomplementarityliterature(Cooper and John 1988). The
derivationof the unconditionalreactionfunctionof the bankis as follows. Let
f3(7C 7e)U~t-l be the continuationpayoffof the bank from t onwards,conditionalonfixed levelsof ir, ir'. Then,Pf(r, ire) is definedby a recursionequation
This
like (7), but for fixed ir, ire; i.e., fl= (+Sfl)(a_Ir+
Ie)2+(1IA_)r2.
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ECONOMICA
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solves to give
(a+ire-)
f(ir, 7')=
+(1
7)L)r2
f3(ir,i-re)+ r'-r=
Substitutingthis into the conditionalreactionfunction(9), and simplifying,we
obtain a quadraticin 7r,
(1 -AS)(a
+ ire)n 2+ [1 -(1
- A)6(a
+ ire)2]1ir-
(a +
ire)
= 0,
which implicitlydefines the unconditionalreaction function (or correspondence), ,F(7re). This always has two real roots for Ae(0, 1). However,one of
these is always negative, which from (9) is inconsistentwith ,B>0. So the
unconditionalreactionfunction is the positive root of the above quadratic,
viewedas a functionof ire. It is easy to show that ir(0)>0, and that ire) is
increasingand convex for ire smallenough,as shown in Figure 1.
The possibility of multiple-feedbackNash equilibriain linear-quadratic
infinitehorizon"3gameshas been noted before,initiallyby Papavassilopoulos
and Olsder(1984). However,they simplyremarkedthat, for a symmetricbut
otherwisequite generalcontinuous-timemodel, the Riccatiequationscould in
principleadmitseveralsolutions:they did not providean explicitexamplewith
multipleequilibria,and in any case our model is not a specialcase of theirs
sinceit is not symmetric.Morerecently,Cohenand Michel(1988)haveproved
uniquenessof feedbackNash equilibriumin a class of two-playercontinuoustime gameswhere:(i) agentshave the same targetvalues (zero) for the state
variable;and (ii) the instantaneouspayoff of any playerdependsonly on (a)
the squareddeviationof the state variableand his own action variablefrom
zero, and (b) the cross-productof the levelsof the two actionvariables,under
the condition that the weight on this cross-productis sufficiently small. Finally,
exampleof a publicfinance
Obstfeld(1991, SectionIV) has a linear-quadratic
model with multipleequilibria;but the model has severalstate variables,and
requiresthat the government'sdiscountrate exceedsthe privatesector's.
How does our resultrelateto this-as yet, ratherinconclusive-literature?
It can be shown (see Section IV(d) below) that multipleequilibriaalso arise
in the continuous-timeversionof our model,so we can makea directcomparison with Cohen and Michel. Our continuous-timemodel differs from the
Cohen-Michel model in that the instantaneous payoff of the union is
-(ir, - re)2; i.e., there is an interactionterm between the union's and the bank's
control variables.(The presenceof this interactionterm is ultimatelydue to
differenttargetsbetweenthe playersfor the state variable.)Multiplyingout
the interactionterm,we see that thereis a cross-productir,ire whichcannotbe
made arbitrarilysmall. In all other respects,our model satisfiesthe Cohenwhich
Michelconditions;consequently,our modelprovidesa counter-example
indicatesthat the Cohen-Michelresultdoes not extendto the case wherethere
between
are non-negligibleinteractionterms(i.e. strategiccomplementarities)
the player'scontrol variablesin the instantaneouspayoffs.
Finally,other relatedmodelsthat generatemultipleinflationequilibriaare
those of Calvo (1988), Brunoand Fischer(1990) and Rogoff (1989). The first
two are public financemodels with non-linearitiesin the governmentbudget
constraintarisingfrom a nonlinearspecificationfor the demandfor money;
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UNEMPLOYMENT DYNAMICS
Rogoff (1989) assumesnon-quadraticpayoffs in a BG type model. Thus, all
these papers generatetheir results from some nonlinearityin preferencesor
constraints,in contrastto our model.
(d) Comparativestatics
The two equilibriaalso possessdifferentcomparative-statics
properties,14which
are summarizedin Table 1. The table indicatesthat the responsesof the low
TABLE1
EFFECTON AN INCREASEIN PARAMETERS
a, 5, A ON INFLATIONRATES If, X, rbg
a
iT
+
+
+
-
-
_
0
i,bg
A
O
+
inflationrate to parametersare both intuitivelyplausibleand consistentwith
those of the inflation rate of the static model, but that those of the highinflationrate are of opposite sign.15It is clear from Figure 1 why this is: any
parameterchangethat raisesthe bank'sreactionfunctionwill increaseir and
reducefr. So the table suggeststhat the bank'sunconditionalreactionfunction
is increasingin a, A, S. This is intuitive;the higherthe persistenceof the unemploymentrate (the highera) or the more the bank caresabout the future(the
higher3), or the higherthe relativeweightthat the bankplaceson employment
versusinflation(the higher 4), the greaterthe incentiveto lower the current
unemploymentrate today, and so the greaterthe incentiveto inflatenow.
Finally,it is worth mentioningthe limitingbehaviourof 7r, iTas a goes to
0 or 1. First, limo
= 'rbg,
lima,o r= oo, so that, as 'insider power', measured
by a, becomesnegligible,inflationper unit of the unemploymentrate tends to
the static level. Of course, as a -+0, the rate u-+O, so actualinflationgoes to
zero. If a = 1, the equilibriumprocess for the unemploymentrate has a unit
root, 16i.e. ut= ut_ l . In this case, if a < p, then there are two inflation equilibria:
if 8 = p there is a unique inflation equilibrium,and if 3> p no equilibrium
exists for a close enough to 1.
(e) The role of the assumption A>O
The role of this assumptionis as follows. In the above analysis, we have
identifiedmultiplepositivesolutionsto the Riccatiequationfor the bank with
multipleequilibria.For this identificationto be valid, it must be the case that,
given a fixed 7Te set by the union, there is a unique solution to the Riccati
equationwhich correspondsto the bank's optimal strategy.For if there are
two solutions to the Riccati equation for a fixed 7Te,only one of them can
correspondto an optimal(and therefore)equilibriumstrategyby the bank.In
this case, we would confuse the non-optimalsolution to the Riccati equation
with an equilibrium.
In this model, it turnsout that A> 0 is in fact sufficientto ensurea unique
solutionto the Riccati equationfor a fixed 7Ce.If 7e is fixed, the bank faces a
standardlinear-quadraticone-persondecisionproblem.In this case, standard
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jointly sufficientconditionsfor a uniquepositivesolutionto the Riccatiequations are the so-calledcontrollabilityand observabilityconditions(Wonham
1979). Controllabilitysimplyrequiresthat the bank can affectu, via ir, and
is satisfiedfrom (5). Observabilityrequiresthat the term in the bank's perperiod payoff, which depends on the state variableu2, has strictly positive
weight,i.e. ) >0.
What goes wrong when =0 is easily shown. The recursionequation of
dynamicprogrammingis, in this case,
- U2_1=max - 2[kU2+ (1 _-A)n+ 6/3U2]
2
if
subjectto ,rc= ru,I , ut =u,_ (a + e - nr).Given 7re exogenous,it is easily
checkedthat the global maximum,conditionalon /, of the right-handside of
this equation (and hence the bank's decision rule) is r = [(A+ ,f)(a + r')]/
(1 +63). Substitutingbackinto the recursionequationgivesa Riccatiequation
326 +p [1- (a + ,re)26] - (a + ,re)22k= 0. As the present value of losses is nonpositive, only non-negativesolutionsto this equationare admissible.If A= 0,
this clearlyhas two non-negativesolutions,one zero and one positive (as long
as e is small enough). However,the optimalpolicy is clearlyto set ir = 0, so
only the zero solution can correspondto the optimalpolicy.
III. SELECTINGEQUILIBRIA
Our model exhibits multiple (two) stable equilibria,one of which Paretodominatesthe other. This raises two relatedquestions.First, can we predict
whichone is more likelyto come about?Second,is thereany way in whichthe
governmentcan ensurethat the good equilibriumis selected?The firstquestion
has been extensivelyconsideredin the game-theoryliterature(e.g. Fudenberg
and Tirole1991).The view of this literatureis thatcostlesspreplaycommunication betweenplayers(so-called'cheaptalk') is not sufficientto ensurethe good
equilibriumoutcome; for example,there is always an equilibriumwhere the
privatesectorrationallyignoresany promisethe governmentmay makeabout
low inflation(or vice versa).
One way out of this, stressedby Krugman(1991) in the context of static
modelswith multipleequilibriaarisingfrom externalities,is to introducecosts
of adjustmentinto agents'decision-making.In his paper,Krugmanshowsthat,
in a simplemodel of trade with externalities,if costs of adjustmentare large
enough, only one of the equilibriacan be reachedfrom any given initialcondition;in otherwords,initialconditions(history)governequilibriumselection.
It is conceivablethat the introductionof, say, costs of adjustmentin inflation,
i.e. a term in (7rt-
Xt_ 1)2
in (8), might alter the Riccati equations so as to
eliminatethe multipleequilibria;but we have not investigatedthis, as it would
involve the analysisof a much more complexmodel with two state variables.
A second approachis to changethe expectationsformationmechanismof
the privatesector;it is well knownthat doing this can changethe set of stable
equilibria.A good exampleis the seignioragemodel of inflationdue to Bruno
and Fischer(1990) and others,which has a continuumof inflationequilibria,
only one of which-the high-inflationequilibrium-is stable under rational
expectations.Brunoand Fischershowedthatintroducingadaptiveexpectations
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reducedthe numberof equilibriato two (with high and low inflation), and
that only the lower of these was stable.17Introducingadaptiveexpectationsin
our modelcertainlyrestoresuniqueness,but for a differentreason.To see this,
take the special case where the private sector fully adapts to past forecast
errors,so that r,e= t_.l. Then, becausethe privatesector is followingan ad
hoc rule, ratherthan choosingexpectations'optimally'given the loss function
-(rt )2/2, the government'soptimal choice of inflation is no longer the
outcomeof a dynamicgame, but is a single-personoptimalcontrol problem,
with a new state variable, (say) st= rt, and state equations st= rt, ut=
aut-I-
7rt+ st -I. It is easy to show that this control problem satisfies the stan-
conditionsfor a uniquesolutionand also for
dardobservability/controllability
a uniquesolutionto the associatedRiccatiequation.(See e.g. Wonham(1979)
for a statementof these conditions.)
A third, and perhapsthe most interesting,approachto selectionis as follows. In our model, the root cause of the multiplicityis the infinitehorizon.
Therefore,whatis requiredto selectthe low-inflationequilibriumis some credible precommitment
on the partof the governmentto a uniquepathfor inflation
startingat some point in the future.The classic exampleis a commitmentto
join a fixed exchangerate regime(say the ERM) at some futuredate, where
the rate of inflationwill be set without regardfor domestic unemployment.
Without loss of generality,suppose that the ERM inflationrate is zero. We
will show that the effectsof such a commitmentwill be: (i) to eliminatethe
high-inflationequilibrium,thus ensuringa uniquetime path for inflation;(ii)
to ensurethat inflationper unit of the unemploymentrate decreasesover time
until entryinto ERM takes place. We also show that, unless the commitment
is credible(i.e. unlessgovernmentintendsto join with probability1), multiple
equilibriacannot be eliminatedin this way.18
The argumentis simple.Supposeat time 0 the governmentintendsto enter
the ERM withprobability1 at dateT. Then,fromT onwards,the presentvalue
of losses to the governmentwill arisefrom unemploymentonly. In general,we
can writethis presentvalue as a linearfunctionof Ut-i, -2JPTUt-_
; from (4),
and the fact that ut= au_ , this will be
-2
00
Z
E
UtS + T=
2)autf/(1-a),
t=O
so PT=l*=Ra2/(l
-a26). Then, over periods 1 to T- 1, optimal inflation
solves the recursiveequation (7) above, where the ,B on the right-handside
and left-handside of (7) are replacedby ft+, and PI respectively.Then, as
before,it is easy to confirmthat the Riccatiequationmust be of the form
(14)
f3t=f(Pt+i)
+ S2f2+ 1+ 2A,pt+,)a2 + SPf+ la2,
= )a2 + (1- )-1(A2
t=,
..
T-1
with terminalconditionPT= P*. Also from (7) modifiedin this way, inflation
at time t = 1, . ., T- 1 will be rt = (A+ 8,t +,)/(I
so the time-path of
inflationpriorto ERM entryis determinedby the sequence{/3I, ... ,JJT}. To
determine the behaviour of {3I, ... ., T}, note thatf(
f)
is a convex quadratic
function,so that thereare at most two roots, ,B< ,, to the equation,B=f(,B);
i.e., the differenceequationhas two equilibria/B,,B.If thereare two suchroots,
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,Bis dynamicallystable startingfrom any terminalvalue PTE[0, p], but P is
alwaysunstable.'9So, it is clearthat, as T-+oo, ,BIwill convergeto ,B.Furthermore, by construction,13*< P, and ,B,,1correspondto the low and high inflation equilibria,i.e., fr= ( + SJ)/(1 -), etc.
All this impliesthe following.Supposethat at date t the governmentmakes
a fully credibleprecommitmentto join the ERM at date t + T, whereT is large.
Then the inflationratejumps immediatelyto a rate 'close' to r. (The longer
T, the closer to 7rthe inflationrate is.) Then, inflationfalls monotonicallyto
XT- I = (, + 6P*)/(1 - ,) in the period before ERM entry. In particular, if the
economy is initiallyat the high-inflationequilibrium,it is effectivelyswitched
to the low-inflationequilibriumby this precommitment.
Now supposethat the governmentcan make only a probabilisticcommitment to ERM membership-specifically,suppose that it can precommitto
joining only with probabilityq in any period.Then the last term on the righthand-sidein equation(8) changesto - 26[(1 - q)JJ+ qf*]ut2. Now the Riccati
equationbecomes
(15)
[(I -
q)2(
a)2]p2+
[6(l
- q)(I + ,)a 2_
l]
+ [+
(I
)qf
]a
,
and the inflation rate in every period, per unit of the unemploymentrate,
becomes
r=
[+ 6((1 - q)/ + q1*)]/(1
-
).
Inspectionof (15) makes it clear that, if (15) has a positive root, it has two
such roots, so that as long as q <1, we still have multipleequilibria.So a
probabilistic,as opposed to deterministic,commitmentto a zero-inflation
regimedoes not eliminatemultipleequilibria,no matterhow high the probability of joining might be.
IV. EXTENSIONS
OF THE MODEL
(a) The union has also a real wage target: 0>0 and p = 0 in (3)
This is the case wherethe union puts some weighton a real-wageobjective,Q,
in (3): empirically,0>0 seems to be plausible(Alogoskoufisand Manning
1988). Then, substitution of (1) in (3) and rearrangementimplies that
the union's effective employment target in period t changes from It to
(l -_o )/(1 + 0).20 The stateequationfor the unemploymentrate,ut, changes
from (5) to:
(16)
ut=n*+a*uti
I(rt
_ Z)
where n* = [(n + Q )0]/(1 + 0) and a* = a/(1 + 0). Equation (16) reduces to (5)
when 0 = 0.
Now in equilibrium,where 7rt= e the rate ut asymptotesto n*/(1 -a*),
which is greaterthan zero as long as Q> -n, becausethen the union's realwage target is consistentwith a level of employmentlower than that of full
employment.In what follows, we assume that Q> -n. This ensuresthat an
increasein 0 has a common-senseinterpretation:namely, a higher 0 means
that the union caresmore for higherwages at the expenseof employment.
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As the employmenttargetsof unionand governmentdifferevenin the limit,
the bank's valuation function at time t is of the form V(u -) = P3o+ flIutlI
-,f2u2 _/2. Makingthe relevantsubstitutionsinto recursionequation(7) and
maximizingwith respectto 1T impliesthat equilibriuminflationis
(17)
irt= ir(Ut- 1: O) = [(+ Spl2)(n* + a*ut-
l)-pI]/(I-D,
which is no longer simply proportionalto the state variable,but involves a
constantterm, PI3.
Substituting(17) back into the analogueof the recursionequation(7) and
equatingcoefficientson ut_ and U2_1, we obtain two equationsin PI, 12:
(18)
(a*)262fJ2+ [,n*a*(
(19
(19),
=~
PI
I3 =
P +
1+ (1 +
,)6(a*)2]P2
+ A,(a *)2 -O
+ 512)(1 + 3f2)
6P2)a* - (1 -)(l
-
6a*)
which have a recursivestructure,with (18) yieldingvalue(s) for /2 and (19)
yieldinga uniquevalue for PI, conditionalon /32 beingknown.Note that (18)
is identical to (12), except that a is replaced by a* =
a/(I
+ 0).
So, concentrat-
ing on the moreplausiblelow-inflationequilibrium,and usingthe comparativestaticsresultsin Table 1 above, we see that 12 < ,Bas long as 0 > 0.
We are now in a position to assess the effect on equilibriuminflationof
increasingthe weight on the wage target, 0, from zero. It is helpfulto begin
with the static BG model. The inflationrates in the static model without and
with wage targetsare, respectively, rbg= A,(n - lu)/( 1 - A) and r* A,(n - 1*)/
(1 - A), where 1*= (lu - OQ)/(1 + 0) is the effective employment target of the
union when the union also has a wage targetof Q. The naturalassumptionto
makeis that Q is consistentwith a loweremploymentlevelthan lu, i.e. Q>>-u;
this is analogousto the assumptionthat Q > -n above. Giventhis, 1*< lu, and
thereforeit is clear that, in the static case, the introductionof a wage target
unambiguouslyincreases equilibriuminflation, through the mechanism of
increasingthe gap between union's and government'semploymenttargets.
However,the inflationbiasper unitof the unemploymentrate is unchangedat
A/(1-
), and hence it is independent of 0.
The effectof an increasein 0 in the dynamiccase is much more subtle,as,
from (17)-(19) above, 0 clearlyaffectsP3Iand ,B2 as well as the time path of
u,. However,we can show the following: as long as 132 is high enough,then,
given an unemploymentrate inheritedfrom the past, ut- I, inflation is lower when
the unionputs a positive weight on a wage target (0 > 0) than when it only cares
about employment (0 = 0). More formally, using the notation introducedin (17),
r(Ut 1: 0) < r(Ut _1 0).21 The intuitionfor this is as follows. 0 > 0 impliesthat
the degreeof persistencein the unemploymentratefalls from a to a * (compare
(16) and (5)). Therefore,a unit reductionin the currentrate will have less
effecton futurerates,which-at a givendiscountrate 6-reduces the incentive
to reducethe currentrate today.
We can now draw on the precedinganalysisto describefairlycompletely
how the timepathof inflationis changedwhen 0 rises.Considertwo economies,
identicalin everyrespect(includingthe sameinitialuo)exceptthat in economy
A 0=0 and in economy B 0>0. Then, from (5) and (16), and the fact that
a* < a, u' for A is initiallyhigherthan uB but, aftersome to, falls below it and
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eventuallytendsto zero, whereasut eventuallytendsto n*/(l - a*). Assuming
that /2 is high enoughthat c(Ut- I: 0) < ,r(ut-1 : 0) (as describedabove), then
there is a t1> to such that ,Ar> 41' for t < t1, but ,rA< 4? for t 2 t1. So, at least
for an initialtime, inflationmay be lower when the union has a wage target.
(b) The union has also an inflation target: 0= 0 and p > 0 in (3)
In this case, the uniondislikesinflation,althoughits trade-offbetweeninflation
and its desiredlevel of employmentmay still differfrom the bank's.If that is
so, the outcome of the game dependscriticallyon whetherthe union takesPt
as given when it chooses w,. The most plausibleis that the union does takePt
as given. This correspondsto the case where there are a large number of
(identical)unionswho do not individuallyperceivethat they can affectmonetarypolicythroughwage-setting(althoughthey obviouslydo in the aggregate).
Then, as 7rtentersin as an additiveconstantin the union'spayoff,it does not
affect wt or , and thus does not change the equilibrium(although union
membersare obviouslyworse off than if p =0). The other case is wherethe
the government.Then, the union takes
union is the Stackelbergleadervis-ad-vis
the governmentreactionfunction(9) above into accountwhen setting1[e. This
case is complexand beyond the scope of this paper.
(c) Stochastic shocks
None of the resultschangein the presenceof stochasticshocks. For instance,
we can introduce a productivity shock, pt=t put- + Zt, 0 < a < 1, where zt is
i.i.d with known variance a2. Then, with E= 1, (1) changes to
+
It =(p,-Wt
1t)-
Now the order of events at any time t is as follows. Insiderschoose wt
beforethe realizationof ut, but knowingall the variablesup to and including
period t - 1. Thus, for insiders,y4 = uptt-I. After the realizationof pt or zt
takes place, the bank and firmschoosept and 1,, respectively,undercomplete
currentinformation.So (5) changesto
ut= aut_I-(rt,-n)-zt.
(20)
The value function of the bank at time t can be written as V(ut I, z)
=
since zt is observable.
(-flut
1/2)
f/2)
IZt)],
(02
However, Et(utzt+l) = 0, so that Ej[ V(ut, zt+ l)] = (-3lu 2/2) - (f32az2/2)where
+ [P 3(Ut-
-
where (u,
Izt) #0
the expectationis takenwith respectto the informationset of the government.
Then we get, for inflationand the unemploymentrate:
-j
(21)
rt
(22)
ut=aut_l-
(autl-
-z)
zt.
Even if the inflationbias and employmentchange because of the supply
shock, the inflation bias per unit of the unemploymentrate, ut, remains
unchangedas in (10) in Section II. The Riccati equations(21) and (22) are
recursive,so that the solution for f3I is again given by (12); then, /2 and P3
follow.
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(d) Continuous time
It is still easy to verify that the continuous-timeversion of the model has
multiple equilibria. The state equation (5) becomes du/dt = (a - 1 - ,r+ ir')u(t),
where ir,
ire
are the feedback coefficients on u(t) chosen by the two players.
The Hamilton-Jacobiequation (the continuous-timeversionof the recursion
equation(7)) is
(23)
rV(u)=max {-O.5pU22+(1-_A),R2]+ J(u)(a- 1-r+,re)u},
ir
whichhas a quadraticsolution, V= -flu2/2. Thenthe equilibriumrateof inflation from (23) is ;r = fl/(1
-
2L), where ,B solves the continuous-time Riccati
equation[132/(1 - X)] - ,[r+2(1 - a)] + X= 0. Again,if this has a positivesolution, it has two positive solutions.
(e) Reputational equilibria
We can also solve our model for what BG call the reputationalequilibrium
(RE), wherea low inflationis maintainedby a crediblethreaton the part of
wage-settersto increasethe growthrate of wagesif the governmentrenegesto
higherinflation.This is done in an earlierversionof the paper(Lockwoodand
Philippopoulos1991); here we report only the main featuresof the RE. In
both the BG model and our model, the RE is characterizedby an incentive
constraintwhich ensuresthat the governmentwill not wish to renegeto the
LMPE
level of inflation.In our model with two equilibria,thereare in fact two such
constraints:one correspondingto each LMPE. Our first-and not terribly
surprising-resultis that the minimumRE inflationrate (perunit of the unemploymentrate) whenthe threatis to revertto the high-inflationLMPEis lower
than the minimumRE inflationrate when the threatis to revertto the lowinflationLMPE. More interestingis the behaviourof these minimalrates as
the influenceof insiders over the union's employmenttarget changes. The
minimumRE when the threatis to revertto the high (low) inflationLMPEis
increasing(decreasing)in insiderpower.22Finally,we can comparetheseminimal inflationrateswith the minimalRE inflationrate in the static BG model,
and identifyconditionsunder which dynamicsmay lead to the possibilityof
lower sustainableinflation.When insiderpower or the discountfactor is low,
it may be possibleto achievea lower inflationthan in the static BG case.
V. CONCLUSIONS
Thispaperhas incorporatedstructuraldynamicsinto the Barro-Gordontheory
of inflationby supposingthat the employmenttargetof wage-settersdepends
on lagged employment.Our first main result is that this can lead to multiple
equilibriumlevelsof inflation,even thoughthe basicmodelis linear-quadratic.
We have also shown that the comparativestatics of the dynamicmodel may
differqualitativelyfrom the static model. In addition, we have analysedthe
implicationsfor inflationof futurestabilizationprogrammesand richerunion
objectives.One interestingextension would be to introducedifferentpolicy
regimes,e.g. to examinehow inflationchangeswhen the governmentcontrols
the interestrate or the exchangerate in an open economy.Anotherwould be
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to introduce several political parties with different preferences over inflation
relative to unemployment. A start on this has been made by Alogoskufis et al.
(1992).
APPENDIX: CONDITIONSFOR POSITIVEREAL ROOTS
For real roots to (12), we require restrictionson the parametersso that P=
(I + p2)(I l
_
)2 - 2(1-
A2)p 0, where 0 < A<1 and O<p=a23<
1. Then, real roots
exist if and only if
(A1)
(Al)
)
(?]U[~
pc-(O,
p]UIj3, ac),
p=
(l__________
(1+i)+2i1/2
whereit follows that p < 1< p. Thus, an equilibriumexists only on the intervalpe [0,
Next, from (12),
p], since p=3a2<1.
(A2)
i,=
(
-)-(I1
+) ) pF Tw1/2
But this holds if and only if
Then, q7?0, if and only if (I-A)-(I+A)pW'T/2>0.
p < (1 - .)/(I + )< 1. This impliesthat an equilibriumexists only on the interval
(A3)
0 <p <po =min {p, (I - A)I(l + A)).
But we can check that p < (I - .)/(1 + A). So, (A3) gives equation(13). U
ACKNOWLEDGMENTS
We would like to thankGeorgeAlogoskoufis,David Begg, OlivierBlanchard,Melvyn
Coles, Jeff Frank, Paul Levine, two anonymousrefereesand seminarparticipantsat
the 1991ASSET Conferencein Athens,in particularCostas Azariadis,for theircomments. All errorsare ours.
NOTES
1. Althoughthe origin of the theorycan be clearlytracedback to the work of Kydlandand
Prescott,it has become known as the 'Barro-Gordontheory of inflation'.We follow the
literatureby adoptingthis terminology,whilestressingthat Kydlandand Prescottwerein fact
the originators.
2. Thispossibilityis interestingbecauseof our linear-quadratic
structure.On the contrary,multiple inflationequilibriaarenot unexpectedin nonlinearmodelsof monetarypolicy,as in Bruno
and Fischer(1990), Calvo (1988) or Rogoff (1989).
3. For example,in Bruno and Fischer (1990) one of the equilibriais always unstable.Also,
multipleequilibriadue to bubblescan be ruledout by a varietyof arguments,sincebubbles
arepossibleonly in dynamicallyinefficienteconomiespopulatedby finitelylivedagents(Blanchard and Fischer 1989, Chapter 5). Again, in models with multiple equilibriadue to
'thin marketexternalities'(Cooperand John 1988,or Pagano 1990), some equilibriacan be
ruledout on the groundsof instability.
4. Brunoand Fischer(1990) get similarcomparative-statics
propertiesof the high-inflationequilibriumin theirpublicfinancemodelof inflation.
5. We can provethatchoosingthe exchangerateor the moneystockdoes not matterfor inflation
and, of course, employmentin a deterministicmodel. In a stochasticmodel, the standard
resultsof Poole hold. However,it would be interestingto find mechanismsthat could break
this regimeequivalence.(We believethat targetzone boundscouldprovidesucha mechanism
by introducingnonlineartermsin the expectationsmechanisms.)
6. A differencebetweenthe asymptoticeffectiveemploymenttarget of the union and 1* can
be generatedwhen we allow the union to have a wage target also. Thus, the analysisof
the case where l*> n is effectivelycoveredby analysisof the cazse with a wage target in
Section IV.
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7. Although under the maintainedassumptionthat l* =n the BG model is not formallya
special case of ours, by focusing on inflationary bias per unit of unemployment,
u, =
n -,
we can make meaningfulcomparisons.In particular,we can show that, as a -.0, inflation
per unit of the gap in the dynamicmodel tends to inflationper unit of the gap in the
static BG model.
8. Note that our definitionof the unemploymentrate,n - lt, only approximatesto the standard
definition(N- L,)/N.
9. Note that the value functionsmustalwaysdependon l, in this way, as the per-periodpayoff
of the bank depends on 1, only through u,.
10. Strictlyspeaking,to ensurethat this is indeed a best response,the sufficienttransversality
condition for the bank must be satisfied,which in this case requirethat lim,,O 8'(8f/b/
9u,1 )u,_,=-lim, .) pStu,2= 0. However,in equilibrium,re =ir, so from (5), u,= au,_, so
conditionlim, --, _iu ,5'= 0 is indeedsatisfiedfor the bank.
that the transversality
11. The first-orderconditionhas the followinginterestinginterpretation.It can be rearrangedto
yield (X+ 8/p) [a - (7r- 7re)]=(- A)r. The right-handside is the marginalcost of an additionalunitof inflation.Theleft-handsideis themarginalbenefitof reducingthe unemployment
rate by one unit in the currentperiod, 4a-(D )], plus the presentvalue of marginal
benefitsfrom reducingthe unemploymentrate by a units in the next period,a2 units in the
periodafterthat, etc; this presentvalueis 8f31a - (7r- re)].
12. Uniquenessrequiresp= p, or 8a2 (+(l
+?--2)1/2)/(l- _.) < 1.
13. In the finite-horizoncase, linear-quadratic
games genericallyhave a uniquefeedbackNash
and Cruz(1979)for continuoustime
equilibrium.This has beenprovedby Papavassilopoulos
and Basarand Olsder(1982) for discretetime. This is confirmedin our model;in the finitehorizoncase,the Ricattiequation(12) becomesa differenceequationwitha terminalcondition
PT+ I=0, whichgeneratesa uniquesequence/, for 0? t ? T. See also SectionIII below.
14. These propertiesare establishedas follows. Considerfirst the high-inflationequilibrium.It
can be shownthat i and henceft alwaysrespondnegatively to increasesin a, s and A: (i) As
p < (1 + .)/(1 - A) from (14) and T 20, (A2) in the Appendixrevelsthat i7 is decreasingin
p, so it follows from (11) that iris also decreasingin p. (ii) A similarargumentappliesfor
A1.Finally,from Figure1, 7rmust have the oppositeresponsefrom ft to a, 8, X.
15. It is interestingthat Brunoand Fischer(1990) and Marcetand Sargent(1989) get a similar
resultin a publicfinancesetting,where,if the economyis at an equilibriumon the wrongside
of the 'inflationtax' Laffercurve,an increasein the deficitleadsto lowerinflation.They call
this equilibriuma 'realtrap'.
16. Note that this cannotbe the case in practice;u, is approximatelyequalto the unemployment
rate,whichis boundedbelow 1 andso cannothavea unitroot.It is truethatu, is bestexplained
as an AR(2) processwith a unit root for many OECD countries(see e.g. Alogoskoufisand
Manning1988),but it is well knownthat unit root tests have low power.
17. In addition,Marcetand Sargent(1989) have shownthat only the low-inflationequilibriumis
stableunderadaptiveleast-squareslearning.
18. A similaruse of a terminalconditionto eliminatemultipleequilibriain the Bruno-Fischer
(1990) modelof inflationhas been suggestedby Bentaland Eckstein(1990).
19. Thiscan be verifiedeitherby drawinga phasediagramfor (14), or by observingthat, asf(-)
is increasingand convex,f(f3) < 1 <f(,B).
20. To see this, note that (1) in (3) gives a minimand [(1 + 9)12] - [21,( u- OKI)]+constant terms,
which obviously has a unique minimum at (lu - O9Q)/(1 + 0).
21. The proof is as follows. Assume that r(u,_1;O)<nf(u,_1;9),
0>0; this requires
1. As a*<a and f2<P this inequality will certainly hold
PI -(X+ 8f2)n*>8(a*,f2-afi)u-
if PI>(.X+8f32)n*>0.This in turn requiresonly a(2L+ fP2)>(1-A)(I -Sa*).
22. At firstsight, this seemsto contradictthe comparativestaticsof the LMPEreportedabove.
However,this can be explainedby observingthat the minimumRE inflationratevariesinversely with the inflationrate used in the 'punishment'of the governmentby the wage-setters.
REFERENCES
ALOGOSKOUFIS, G.
and MANNING,A. (1988). Unemploymentpersistence.Economic Policy, 7,
427-69.
and
LOCKWOOD, B. and PHILIPPOPOULOS,A. (1992).Wageinflation,electoraluncertainty,
the exchangerate regime;theoryand UK evidence.Economic Journal, 102, 1370-95.
ALoGOSKOUFIs, G. and MANNING,A. (1988). Unemploymentpersistence.Economic Policy, 7,
427-69.
R. and GORDON, D. (1983a). Rules, discretion and reputation in a model of monetary
policy.Journal of Monetary Economics, 12, 101-21.
BARRo,
( The LondonSchoolof Economicsand PoliticalScience1994
76
ECONOMICA
[FEBRUARY
BARRO,R. and GORDON,D. (1983b). A positive theory of monetary policy in a natural-rate
model. Journal of Political Economy, 91, 589-610.
BASAR,T. and OLSDER,G. (1982). Dynamic Noncooperative Game Theory. London: Academic
Press.
BENTAL,B. and ECKSTEIN,Z. (1990). The dynamics of inflation with constant deficit under
expected regime change. Economic Journal, 100, 1245-60.
BLANCHARD,0. and FISCHER,S. (1989). Lectures on Macroeconomics. Cambridge,Mass.: MIT
Press.Mass.
I
and SUMMERS,L. (1986). Hysteresis and the European unemployment problem In S. Fischer
(ed.), NBER Macroeconomic Annual. Cambridge,Mass: MIT Press.
BLANCHFLOWER, D., OSWALD,A. and GARRET,M. (1990). Insider power in wage determination.
Economica, 57, 143-70.
BOOTH, A. (1983). A reconsideration of trade union growth in the United Kingdom. British
Journal of Industrial Relations, 21, 377-91.
BRUNO, M. and FISCHER, S. (1990). Seigniorage, operating rules and the high inflation trap.
Quarterly Journal of Economics, 105, 353-74.
CALVO, G. (1988). Servicing the public debt. American Economic Review, 78, 647-61.
CANZONERI, M. (1985). Monetary policy games and the role of private information. American
Economic Review, 75, 1056-70.
CARRUTH, A. and OSWALD,A. (1987). On unionpreferences
and labourmarketmodels:insiders
and outsiders.Economic Journal, 97, 431-45.
COHEN, D. and MICHEL,P. (1988). How should control theory be used to calculatea timeconsistentgovernmentpolicy?Review of Economic Studies, 55, 263-74.
COOPER, R. and JOHN, A. (1988). Coordinating coordination problems in Keynesian models,
Quarterly Journal of Economics, 103, 441-63.
FUDENBERG, D. and TIROLE,J. (1991). Game Theory. Cambridge, Mass: MIT Press.
HORN,H. and PERSSON,T. (1988). Exchange rate policy, wage formation and credibility. European
Economic Review, 32, 1621-36.
KIDD, D. and OSWALD,A. (1987). A dynamic mode of trade union behaviour. Economica, 54,
355-65.
KRUGMAN,P. (1991). History versus expectations.QuarterlyJournal of Economics, 106, 651-67
KYDLAND,F. andPRESCOTT, E. (1977).Rulesratherthandiscretion:the inconsistencyof optimal
plans.Journal of Political Economy, 85, 473-92.
LAYARD, R. and BEAN, C. (1989). Why does unemployment persist? Scandinavian Journal of
Economics, 91, 371-96.
LEVINE,P. (1988). Does time consistency matter? Centre for Economic Policy Research Discussion
Paper no. 227.
and insider-outsiderelaLINDBECK, A. and SNOWER,D. (1986). Wage-setting,unemployment
tions. American Economic Review, 76, 235-9.
(1987). Union activity, unemploymentpersistenceand wage-employmentratchets.
European Economic Review, 31, 157-67.
LOCKWOOD, B. and MANNING,
A. (1989). Dynamicwage-employmentbargainingwith employ-
ment adjustment costs. Economic Journal, 99, 1143-58.
and PHILIPPOPOULOS,A. (1991).Insiderpower,employmentdynamicsandmultipleinflation
equilibria. University of Essex Discussion Paper no. 378.
MARCET,A. and SARGENT,T. (1989). Least-squares learning and the dynamics of hyperinflation.
In W. Barnett, J. Geweke and K. Shell (eds.), Economic Complexity: Chaos, Sunspots, Bubbles,
and Non-linearity. Cambridge University Press.
MILLER,M. and SALMON,M. (1985). Policycoordinationand
dynamicgames.In W. Buiter and
R Marston(eds.),InternationalEconomic Policy and Coordination.Cambridge University Press.
NICKELL, S. (1990). Unemployment:a survey.Economic Journal, 100, 391-439.
OBSTFELD,M. (1991). Dynamicseignioragetheory:an exploration.Centrefor EconomicPolicy
ResearchDiscussionPaperno. 519.
OUDIz, G. and SACHS,J. (1985). International policy coordination in dynamic macroeconomic
models. In W. Buiter and R. Marston (eds.), International Economic Policy and Coordination.
Cambridge University Press.
PAGANO, M. (1990). Imperfect competition, underemployment equilibria and fiscal policy.
Economic Journal, 100, 440-63.
(D The London School of Economics and Political Science 1994
1994]
UNEMPLOYMENT
DYNAMICS
77
PAPAVASSILOPOULOS,
G. and CRUZ, G. (1979). On the uniqueness of Nash strategies for a class
of analytic differential games. Journal of Optimization Theory and Applications, 27, 309-14.
and OLSDER,G. (1984). On the linear-quadratic,closed-loop, no-memory Nash game. Journal
of Optimization Theory and Applications, 42, 551-60.
ROGOFF,K. (1989). Reputation, coordination and monetary policy. In R. Barro (ed.), Modern
Business Cycle Theory. Oxford: Basil Blackwell.
SARGENT, T. (1987). Dynamic Macroeconomic Theory. Cambridge, Mass.: Harvard University
Press.
TABELLINI,
G. (1988). Centralized wage setting and monetary policy in a reputational equilibrium.
Journal of Money, Credit and Banking, 20, 102-18.
WONHAM, W. (1979). Linear Multivariable Control: A Geometric Approach. Berlin: SpringerVerlag.
? The London School of Economics and Political Science 1994