N~
90
RATIONAL EXPECTATIONS, INCOME POLICIES ANO
GAME THEORY
Mario Henrique Simonsen
.
RATIONAL EXPECTATIONS, INCOME POLICIES AND GAME
Ma~io
Hen~ique
THEORY
Simon~en
1- Strategic Interdependence and Income Policies
That aggregate demand discipline
is
a
necessary
condition for sustained price stability has long been known by
economists and well advised policy-makers.
Yet it may not
sufficient to successfully stop a big inflation, as
.
overlooked inflationary inertia, thus leading
sta:gflation.
evidenced
programs
by the failure of a number of IMF-supported
to
be
that
dismal
Not surprisingly, countries like Argentina,Israel
and Brazil recently decided to focus on the .supply
side
inflation, attempting to stabilize prices by combining
policies with monetary reforms.
of
income
Whether these experiments will
yield success stories or not depends on a number
of factors,
including an adequate aggregate demand management. 'lhe interesting
issue is that they were·inspired on
a
sound
approach to the supply side of inflation.
argue in the following
disc~ssion,
income
game theoretical
In fact, as we shall
policies
necessary to coordinate individual behavior according
rational expectations models.
may
be
to
Moreover, monetary reforms
be used to dissipate the Fischer-Taylor inertia
that
may
is
consistent with the rational expectations hypothesis.
The central
quest~on
is to understand what
inertial inflation and how income policies
can
break
dependence of the inflation rate on its past behavior.
traditional explanation of the late sixties, combining
--~--
- - -
causes
the
The
the
.2.
natural unemployment rate hypothesis wi th adaptati ve expectations,
was a benchmark in terms of interpreting inflation as an autoregressive process, although it never made clear how
policies might affect expectations.
income
In any case, it was
soon
eclipsed by the rational expectations revolution that dismissed
adaptative expectations as "ad-hoc" assumptions usually
inconsistent with optimal decision-making.
Under the aegis of the new classic economics,inertial
inflation could only. spring from two sources, auto-regressive
expected rates of rnonetary expansion, and outstanding contracts
based on previous inflationary expectations.
The first source
could be dried up by a credible monetary rule, once money supply
was put under the control of respectable and independent Central
Bankers.
The second source could only be responsible
ternporary and dampened inertia.
for
Hence, painless inflation cure
appeared as a strong possibility, perhaps
not
in
treatrnent , but at least in a gradualistic approach
a
shock
to
stability, in line with the rnaturity of old contracts.
price
Since
the length of non-indexed contracts is a decreasing function of
the inflation rate, the transition from high to zero inflation
could be quite rapid.
For instance, if rnoney wages were reset
once a year, adjustrnent dates for different labor groups being
uniformely distributed over time, inflation rates could
be
brought down from 20% in year O to 9% in year I and to zero
in
year 2 with no recession at all.
Anti-inflationary policies in the early eighties were
painful enouqh to suggest that inflation rates might be
held
back by a force ignored in rational expectations models,namely,
inertia caused by strategic interdependence arnong privateeconomic
.3.
agents.
In fact, rational individual behavior, besides depending
on anticipated economic
decisions.
policies~
also depends on other individuals'
Moreover, economic agents must make
their
own
act,
wnich
decisions without knowing how other players will
brings inflation theory into the field of games with imperfect
information.
Rational expectations literature explicitly
recognizes the problems of strategic interdependence between the
private sector and the government, activism meaning
that
government decides to act as a dominant Stackelberg
player.
But, as noted by Tobin, this plainly ignores that,
auction markets, there is no sense in treating
sector as a single player.
the
Summing up, the central
the
except
in
private
weakness
of the rational expectations macroeconomics is to describe as a
two-person game what actually should be analysed as
person game.
Tobin correctly guessed that provided
an
n+l
price--setting
games were appropriately discussed in terms of strategic
interdependence, inertia would emerge as a powerful inflationay
force.
The same -idea, incidentally, had
been
grasped
by
Modigliani in his classic contention as to the use of the rational
expectations hypothesis in non-auction markets.
What kept alive
the academic prestige of the new classic economics was that none
of these criticisms was explicitly formalized in terms of
game
theory.
The central question to be discussed is:
what
is
rational decision doing in a non-cooperative n-person game with
imperfect information?
The new classic economics assumption is
that alI players should hit a Nash equilibrium in the first move.
The new Keynesian school simply rejects such a narrow concept
of rationality.
In fact, the choice of a Nash strategy
only
.4.
means rational behavior if each player can be assured that alI
other players will choose consistent Nash strategies.
In
the
absence of such a guarantee, it may be nothing but naively
imprudent
pla~ing.
Cautious economic agents might well prefer
to start the game with the defensive maxmin strategy, which
yields maximum pay-off in the worst conceivable scenario.
In a
few games, such as the prisoners' dilemma and zero-sum two-person
games, Nash equilibria present strong stability properties.
Except that, in alI these cases, they are the outcome of maxmin
strategic' choices.
Wage-price setting belongs to a much
more
complicated class of games, as will be shown in section 2, where
rationality cannot be so narrowly defined.
Rational expectations literature .uses two alternative
approaches to by-pass these game theoretical complications. One
is to bring on stage a lValrasian auctioneer that prevents
any
transaction as long as any player can improve his pay-off given
the strategic choices of the remaining participants.
The other
is to assume that alI available information is common knowledge,
each player taking into account the reaction functions of other
players before announcing his 'final decision.
That the outcome
is a Nash equilibrium should not be received as a surprise.
fact, it is a Nash equilibrium by definition.
The problem
that even in auction markets there is some trading
out
equilibrium, so that the Walrasian
only
auctioneer
accepted as a first-order approximation.
can
is
of
be
As to non-auction
markets, the idea that a Nash equilibrium is hit at the
move as a result of a mental exercise involves
slips.
In
some
First, to engage in such a mental exercise,
economic agent must be convinced that the others will
first
serious
every
do the
same, otherwise it will be apure wasmof time and money.
.5.
Second, when the mental exercise is feasible, namely, in a game
with a small number of participants, hitting a Nash equilibrium
may be a poor achievement. Why not try Stackelberg dominance,
or, in the case of repeated games, use signalling and threats
to produce cooperation?
Third, with a large number os players,
each ignoring the other participants' pay-offs, the
exercise
may be impossible.
Moving back to our leit-motif, let us assume that after
prolonged inflation the Central Bank announces that
it
stop printing money
unc~anged.·
$0
as to keep the nominal G.N.P.
will
Even if the general perception is that the nominal G.N.P. will
be immediately stabilized by a new monetary constitution
(an
extreme hypothesis, since monetary austerity is not usually
backed by a constitutional reform), prudent price
setters
should not take the lead in stO?Ping price increases, as
long
as they consider the possibility of further price increases
other sectors.
In fact, the first to jump has little
in
to gain
if he is followedOby the remaining participants in the game, and
much to lose if heOjumps alone.
In the latter case the penalty
is concrete, a real income cut, while the reward is abstract:
additional customers' orders that cannot be met since
has only 24 hours.
the
day
The fact that excess demand does not mean
additional effective demand nor additional income
helps
to
why prudent price-setters should stick to someti1ins
such as a maxrnin strategy.
Of course, in a repeated game with a large number
of
small players, leaving virtually no space for coalitions, threats,
signalling or Stackelberg dominance, maxmin strategies that do
not yield a Nash equilibrium are not likely to be indefinitely
.6.
repeated.
The price-setting model of section 2
indicates how
prudent players in a repeated game may gradually narrow
the
conceivable range of other participants' strategies,triggering
a "tatSnnement approach" to a Nash equilibrium.
Little can be
said, however, about convergence speeds, which may be painfully
slow after a prolonged period of high inflation rates.
The foregoing discussion provides the rationale
income policies: Governments should play the
Walrasian
role
auctioneer, speeding up the location
of
of
invisible hand.
the
Nash
equilibria, namely, using the visible hand to achieve
rational expectations models assume to be the
for
role
what
of
the
It should be stressed that the central function
of incomes policies is not to constrain individual decision
making, but to clear externalities in a garoe with imperfect
information, telling each actor how the others will play. This,
incidentally, dismisses a traditional argument against income
policies, i.e. that governments are not better equipped
than
private markets to identify individual Nash strategies. In fact,
the central problem is not to discover such strategies, but
coordinate their simultaneous playing.
to
This also explains why,
in a second stage of the stabilization program, wage-price
controls should be removed gradually, at successive sectorial
steps, and not in the one-shot manner.
In fact, everi if income
policies are successful enough to bring the economy to a Nash
equilibrium, there is no way to convey such information to the
participants of the game.
Each player finds himself in
equilibrium, but does not know if the same applies
players.
to
other
If controls are lifted sector by sector, players will
realize that no participant in the game will increase
prices
.7.
even when allowed to do so.
On the other hand
theone-shot
approach would simply bring back the uncertainty of
players as to what other players will do, perhaps
individual
triggering
large defensive wage-price increases.
Of course, the chances Df hitting a Nash equilibrium
through income policies are extremely remote, and the fact that
wage-price controls yield some supply shortages should not come
as a surprise.
The central question is what is worse in terms
of social welfare, a few product shortages, that eventually may
be -overcome."
·by··· ·..imports, or
unemployment,
massive
which is notlüng but a shortage of jobs?
From this point
view, objections to income policies should not be taken
of
too
seriously, especially when the problem is to fight a big inflation
with strong inertial roots.
This
j~
alI the more
income policies can be managed with appropriate
so
because
flexibility,
substituting price administration for price freezes.
A more fundamental contention is that the
temporary
success of income policies may lead policy-makers to ignore that
price stability cannot be sustained without aggregate
discipline.
demand
That such misleading signals are a true risk has
been known since the times of the Roman Emperor Diocletian, and
the list of income policy failures is too long to be neglected.
Yet the converse is also true.
Trying to fight a big inflation
only from the demand side may lead to such dismal
stagflation
that policy-makers may well conclude that life with inflation is
preferable to life with a monetarist stabilization programo
.8.
2- A Price-setting Game
Let us assume an economy with a continuum of
goods,
each one produced by an individual price-setter, and where the
nominal outuput R is controlled by the Government(l). The nominal
output is preannounced, but prices must be set simultaneously,
each agent ignoring how the others will act.
Production starts
after prices have been set, according to consumers' orders.
All individuals have the same utility funccion:
U
x
= Lbx
]
10
(0<:
a<
where Lx stands for leisure time and qxy
good Y by individual x
~
(O
x
~
1) .
1
(l)
; b >0)
for the consumption of
The supply S
x
of good
x
equals the number of daily working hours of individual x:
=
(2)
24 - Lx
hence, indica ting by P Y the pr ice of good y (O ~ y ~
1) ,individual
budge constraints are expressed by:
=
R
x
(3)
where R is the nominal income of economic agent x.
x
Constrained utili ty maximization yields the demand Qy for
good y:
(4)
(1) In line with the monetarist tradition, one may assume that the Central
Bank controls some monetary aggregate M that determines the nominal
n
output.
.9.
where:
R =
m
=
J:
(5)
a
l-a
(6 )
and where the consumer orice index ~ is determined by:
p-m
dx
. x
(7 )
Let us first examine the conditions for market clearing. 5ince the quantity sold of good (x cannot exceed Qx :
,.
{ P 5
x x
(8)
Individual's x utility will then be expressed by:
R
P
x
(9)
(24-5 )b
x
Once all ~rices have been set, Ux is maximized for:
24
l+b }
(10)
since there is no incentive to increase supply either beyond
demand or beyond the point that maximizes 5 x (24-S x )b. Hence alI
markets are cleared if and only if:
24
(O ~
x
~ 1)
l+b
.
I
.10.
or, equivalently, if and only if the inequality:
l+b
24
holds for every
(11 )
x E [0;1 ]
Let us nON determine tre optimun
price-setting ruI e under
perfect foresight. Each economic agent should choose P x so
to maximize his or her utility U
x
as
given R and P. Market clearing
can be taken"for granted, since individuaIs are encouraged
increase prices not only to the point of eliminating excess
to
d~,
but also to take advantage of their standing as monopolistic
competitors. Hence, P
p
x
should be chosen so as to maximize:
O
x-x
(24-Q )b
x
P
Taking into account demand equation (4), the optimum
price setting rule yields:
Px
m+l
=
=
m+l
l+b'
24
(12)
b
(13 )
where:
b'
m
and leading to:
1
U max
x
={
24
l+b'
24b,}b
{ l+b'
{_R }
P
m+l
(14)
Combining (12) and (7), the ootimum ?rice-setting rule
·11.
under perfect
p
foresight is expressed by:
=
x
l+b'
24
(15)
R
This corresponds to the unique Nash eguilibrium of the
price-setting game. lt should be noted that equation (15) describes
a monopolistic competition equilibrium, since each good is produced
by one single individual. lf alI producers behaved as competitive
x
price-takers, the supply of good
would maximize U
.
x
given the
price system. Since:
p S
x x
p
(24-S )b
x
the competitive supply of good
sx =
x
would be expressed by:
24
1+b
Introducing equations (4) and (7), competitive equilibrium prices
would be given by:
p'
x
=
l+b
24
R
The higher m, the lower the ratio between monopo1istic
competition and pure1y competitive prices, approaching 1 as
tends to infinity. This is to say that
m
m
indicates the degree
of competitiveness in the economy.
Let us now turn to the dynamics of our price-setting
game. We shallassume that for t ~ O our economy has adjusted for
chronic inf1ation at a constant rate r per unit of time. This is
·12.
to say that the government expands the nominal output according
to Rt
=
(l+r)R _
t l
and that price-setters have already located
.
.
the Nash equilibrium path according to equation (15), which also
yields P
t
=
(l+r)p _ . Suddenlya new administration steps in
t l
and decides to stabilize the nominal output, announcing Rt
for t
~
=
Ro
1.
According to rational expectations, if
new
the
administration presents a credible stabilization program, inflation
will stop immedia tely., wi th no side effects on output.
This
results from two assumtions:
i) equation (1) and (2), which describe individual
preferences, endowments and production sets, are common knowledge.
Hence, all their mathematical consequences, including equation
(15), are also common knowledge;
ii) since all individuals believe that Rt
t
~
=
Ro' for
1, price-setting according to equation (15) will lead to
immediate price stabilization w'ith no side effects on output.
Now, two reasons make these assumptions highly
questionable:
i) even if equationS (1) and (2) are common knowledge
and if alI individuaIs believe that the government will actually
stabilize the nominal output, this is ntt enough to convince
price~setters
to stop any further price increase, To use equation
(15), each player must also be convinced that alI other players
do believe. that the government will
i~nediately
stabilize
the
nominal output. This is to say, credibility must also become
common knovlledge;
ii) to assume that all preferences, endowments and
oroduction sets are common .knowledge obviously sounds like
.13.
science fiction. A more plausible assuption restricts individual
knowledge to his own preferences,to production possibilities
and to the demand function for his product. This is to say,
individual
x
is well informed enough to derive the optimum
equation (12) but not to solve the general equilibrium model that
leads to equation (15).
Once these objections are taken into account, one must
conclude that even if alI individuaIs believe that the government
will actually stabilize the nominal output, the only concrete
information about the general price leveI in period 1 is
it must lie in some point of the interval P
o
~
P
1
~
that
P (l+r) .How
o
to start the price-setting game at this point is a question that
bears no single answer in terms of rational behavior.
Each P
I
in the interval [po' po(l+r)] is a possible
state of nature, and uncertainty may be faced through at least
three different approaches:
a) IndividuaIs may attribute subjective probabilities
to the different states of nature and then set prices so as to
maximize the expected utilitYi
b) individuaIs may feel unable to attribute subjective
probabilities to the different states of nature and prudently
choose the pure maxmin strategYi
c) individuaIs may treat the problem as a game againit
nature and play the saddle-poit mixed strategy.
We shall concentrate on approach b) which avoids the
random choices of mixed strategies, as well as the need to create
probability distributions without any concrete foundation, and
which simplifies the mathematics of inflationary inertia. It
should be stressed, however, that either of the other two
.14.
approaches also leads to inertial problems. In the whole discussion
we sha1l assume that each individual actually believes
Rt
= Ro
>
for t
that
1, and that the government does stabilize
the::
nominal output.
Equation (14)
indicates that, given the nominaloutput
R,the higher the general price level P, the 1m-ler the maximum
utility that the individual can reach by an appropriate
price
setting. Hence, the pure maxmin strategy for individual x is to
set P
x
according to equation (12) w1th the general price 1evel
at its upper bound P max
P m+l
=
x
l+b'
24
RP m
max
(16)
The observed consumer price index, according
to
equation (7), wi1l then be exoressed by:
p
m+l
=
l+b'
24
(17)
In our specific cas.e, since -max
P
P
o
=
l+b'
24
= po(l+r)
and since
R :
o
= P o (l+r)
m
m+l
which shows that prices continue to increase in period 1 in spite
of the credible.nominal output stabilization. The counterpart, of
course, 1s a recession, with real output falling below the fuil
employment level. To further describe the dynamics of inflation,
an additional hypothesis is needed on how P max is estimated for
t ~ 2.
A possible revision rule, assuming R to be kept unchanged,
.15.
is provided by:
(P max)
t
(18 )
=
which assumes that, as a result of nominal output stabilization,
economic agents estimate a non-increasing path for the inflation
rate. Combining equations (17) and (18), taking into account that
p
o
=
l+b'
24
Ro' and making P
t
=
be
ln P t ,price dvnamics will
described by the difference equation:
(19 )
where Pt converges to the Nash equilibrium Po' bu·t 'vlhere li ttle
can be said about convergence speeds.
Equation (19) a?parently
suggests that the higher m, the lower the convergence velocity
to the Nash equil1.briumi a striking conclusion, since in our rrodel
a high m
m~ans increased competition among price-setters.
Yet
equation (4) shows why this conclusion is not to be taken
seriously.
In fact, the higher m, the higher
the
costs
misestimating the consumer price index P, encouraging
price resetting.
of
faster
Summing up, the time unit inteval in equations
(18) and (19) is also a decreasing function of m,
50
that
no
immediate relation can be found between the degree of
competitiveness and inflationary inertia.
Equation (19) invites us to revisit the old-fashioned
hypothesis of
adaptative expectations.
The traditional
of the fifties and sixties was obviously naive, in that
version
it
ignored the obvious fact that expectations respond to changes
.16.
in economic policy.
Yet it implicitly embodied
some
game
theoretical common sense that has been completely overlooked by
the rational expectations revolution.
Of course,
no
simple
backward-Iooking rule su6h as equation (19) can be taken as
satisfactory description of how expectations are formed.
does
~erve,
a
It
however, to recall an essential point, which is"that
thereois no reason why in an n-person non-cooperati ve game
Nash equilibrium should be hit at
the first move.
a
Economic
agents may be alI forward-Iooking, but this does not make them
naive enough not to know that uncertainties about other people's
behavior are an essential ingredient in decision-making.
Summing up, credibility in rational expectations
macroeconomics is a three-storeyd building.
The first story, a
strenuous but perhaps feasible construction, is a commitment to
nominal output stabilization beyond any individual doubt.
second, perhaps only attainable through a
mass communication media, is to convince
The
-dictatorial use of
everybody
that
everybody has already been convinced that nominal output will
be effectively stabilized.
The third, which requires a formidable
extension of common knowledge, is to teach price-setters
that
they can no longer benefit from further price increases. Rational
expectations macroeconomics assumes that, once the first story
has been built, the invisible hand will automatically build on
the other two.
The foregoing analysis indicates to what extent
strategic interdependence brakes the power of the invisible hand.
The potential role for income policies is precisely to
build up the upper storeys of the rational expectations
credibility modelo
Once the first storey has been completed,
the other two may be immediately built on by a
~rice
freeze that
.17.
convinces alI economic agents to take P
max
= Po
~
for t
preventing the we1fare losses of an interim recession.
1,
Game
theory here te11s us the true story behind the vague idea that
income policies mayreverse inflationary expectations.
Their
actual role is to coordinate the simultaneous playing of
Nash
strategies.
The obvious danger is the temptation to build
upper floors before having completed the first.
the
In our model,
this would mean freezing Po without preventing further expansion
of the nominal output R. Initially
th~
economy might experience
a euphoric expansion in consumption, employment and output,
long as the:.. increase in nominal incomes was matched by a
as
cut
in
Then, in line with the long list
of
income policy failures, the program would collapse because
of
monopoly profit margins.
generalized shortages.
In the preceding model general shortages
would emerge if the nominal output were expanded beyond:
=
l+b '
l+b
(20)
the right side above corresponding to the
ratio
between
monopo1istic competition and perfectly competi tive equilibrium
prices.
The fact that after the price freeze some expansion in
nominal output is possible without yielding supoly shortages
helps to explain why price controls .are usually so successful
in the very short run.
Price-setters are converted into price-
takers, thus being forced to accept some squeeze in their' profit
margins.
As long as the latter do not falL below perfect
competition margins, nominal output expansion leads
output growth and increased welfare.
to
both
Yet this eu?horic start
.18.
is nothing but an income policy trapo
In fact,
once
price
controls are lifted, producers will restore their previous margins
by increasing prices and reducing quantities.
Moreover, an
ominous possibility is that policy-makers may misinterpret market
signals and become convinced that, once prices have been frozen,
growth can be prornoted by relentless money printing.
3- Staggered Nage Setting and Inflationary Inertia
Let us now examine the complications created
staggered wage setting.
by
In the following discussion while labor
is assumed to be homogeneous, workers are uniformly
distributed
among a continuum of classes, one for each real number O
~
x
<
1.
Nominal wages of class x are reset at time x+n, for every
positive, zero or negative integer n.
This is to say
individual nominal wages move by steps of time length
that
T=l (the
step curve velow S(T} in figure 1), but that adjustment
dates
for different classes are uniformly spread over time.
Under the above assumptions, the average nominal wage
ar time
t
will be given by the shaded area in figure 2:
w(t)
=
(t
S(T) dT
(21 )
)t-l
S(T) indicating the individual nominal wage of the class with
resetting date
T (t-l
~
T
<
t).
To make sure that the integral
on the rigth side of equation (21) does exist, we shall
S(T) to be a continuous function of T.
assume
S Cr)
S (T)
x
figure
x+l
figure
I
2
Let us further assume that the price leveI Q(t)
is
determined by multiplying unit labor costs by a constant mark-up
factor:
=
Q(t)
(l+m)bW(t)
where m
is the profit margin and
b
the labor input per
output.
In the following discussion we shall overlook
shocks as well as both cyclical and long-term changes
productivity, thus treating b
as a constant.
unit
supply
in
This makes
the
average real wage W(t)/Q(t) a constant:
W(t)
where
z-l
=
=
(22)
zQ(t)
(l+m)b.
Staggered wage setting in an inflationary economy
fntroduces two different real wage concepts, the peak
average.The peak k(T)=S(T)/Q(-r)
and the
(OP in figure 3) is the individual
purchasing power of the class with resetting date T, immediately
after the nominal wage increase.
The average:
.20.
Q (T )
Q(o)
do
indicated by the shaded area in figure 3, is the worker's average
real wage in the period where his nominal wage remains fixed. For
Q(O)e
TIT
Q' (o)/Q(o) is a constant
TI
example, if Q(o)
=
that is, if the inflation rate
,
the peak/average ratio will be given
,
by:
k(T)/Z(T)
=
(24 )
l-e
an increasing function of
TI
-TI
that, for small inflation rates, can
be approximated by
k(T)/Z(T)
=
I
TI
+
-2-
O-T
I
figure 3
what really matters for both employers and employees
is the average real wage Z(T)., Yet the only objective element in
each wage contract is the real wage peak k(T). That the peak
should be set so as to achieve a target average Z(T) seems pretty
obviUQs. Under perfect foresight one may plausibly assume
fUND,\ÇAO GF.TPUO VARGAS
DU01 ECA M·\ 'UI IltF'inl()II"~ SIMO"iSf:.
that
.21.
k(T) is calculated so as to make Z(T)=Z
in
case
of
full
<
employment, unemployment forcing workers to accept Z(T)
and excess demand for labor leading to
Z(T)
>
Z
z. Complications
arise when strategic interdependence is brought on stage.
In
fact, the optimum k(T) depends on both the nominal output path
J.
and the cost of living curve in the time interval [T, T + 1
Even if the former is credibly announced by the
government,the
problem remains that the cost of living path will depend
how other labor groups will react in terms of nominal
increases.
on
wage
In fact, according to equations (21) and (22):
ZQ' (t) =
w'
(t) =
(25)
S(t)-S(t-l)
Let us first explore the mathematics of staggered
wage setting.
Three basic theorems are proved in the Appendix:
Theorem 1 (The constant inflation rate theorem):
for t ~ 0, Q(t) = Q(O)e
TIt
If,
, where TI is a positive constant, then
Z(T) converges to Z and the peak/average ratio to:
lim
k(T)/Z(T)
=
T+OO
1T
l-e
-1T
Theorem 2 (The peak/average ratio theorem). lf,
alI t
~
0, k(t) = rz, r
for
indicating a constant greater than one,
then the inflation rate converges to TI, where:
r
=
1T
l-e- TI
Theorem 3 (The non-neutrality theorem): If the
rate decreases for t-l ~ T
<
t+l, then Z(T.)
>
Z
for
inflation
some
.22.
t-l~ T~
t.
The first theorem explains why chronic inflation
eventually leads to formal or informal indexation.
lf
prices.
increase at approximately stable rates, real wage peaks will be
recomposed at fixed time intervals.
In practice this is achieved
by periodic nominal wage increases for cost of living escalation:
S(T) = S(T-l)
making k(T)=k(T-l).
The second theorem proves the converse.
Since
indexation periodically resets real wage peaks above the
real
average the economy can afford to pay, inflation becomes
predominantly inertial, being no longer the result of too much
money chasing a few goods.
The problem at this stage
is
no
longer to explain why prices actually increase, but to explain
how inflation can accelerate or decelerate.
Excess demand for labor is one possible source
inflation acceleration by making k(T)
>
k(T-l), namely,
substituting indexation plus for simple cost
escalation.
~reased
of
of
by
living
inflationary expectations can
do
the
same, either because economic agents anticipate more expansive
monetary policies or because each labor group expects
other
labor groups to move from simple indexation to indexation plus.
Adverse supply shocks, lowering the equilibrium real wage aV2rage
z, are another cause of inflation acceleration in an
economy.
Recomposing the same real wage peaks
increasing the coefficient
equilibrium inflation rate.
r
now
indexed
means
in Theorem 2, thus lifting
the
Finally, if labor unions manage to
.23.
reduce the wage adjustment interval, say, from twelve
months, what was previously the annual inflation
to
rate
six
will
become the six-month cost of living increase, as occurred
Brazil in late 1979.
This, once again,
~s
in
a consequence
of
the peak/average ratio theorem, where TI is the equilibrium
inflation rate for the wage resetting time intervalo
One might argue that the foregoing analysis averstates
the problem of inflationary inertia by ignoring that the length
of nominal wage contracts is a decreasing
inflation rate.
function
of
the
And that, once this fact is taken into
consideration, a wage escalation clause is not a serious
obstacle to fight inflation.
A virtuous circle can be put
in
~n
motion by a slight decrease in the inflation rate, which
turn will increase the length of the nominal wage contracts,
further lowering the inflation rate and so on.
The
essence of
the argument is that wage indexation at constant time intervals
is not a convincing arrangement, since it makes average
wages a
decrasi~g
function of the inflation rate, a
real
superior
scheme being the trigger-point system where wages are adjusted
at fixed inflation rate thresholds.
Both trigger-point and fixed time interval indexation
schemes are found in inflationary economies, and one may argue
that the second arrangement is nothing but a practical
implementation of some implicit trigger-point, since
wage
adjustment intervals usually decrease as the inflation rate
jumps up.Two points, however, should be stressed.
First, once
annual inflation rates reach a high two-digit figure or
more,
trigger points become highly impractical, since nominal
wages
can be adjusted monthly, quarterly or every six months,
but
.-------------------_.---.
------------
.24.
not at time intervals such as two months, seventeen
three hours.
days
and
Second, the response of the nominal wage contract
lenght to changes in the inflation rate tends to be highly
asymmetric.
rf inflation accelerates, employers and employees
may naturally agree on shorter indexation periods,
leading
to
further inflation acceleration, and paving the road to
hyperinflation.
Yet, once inflation falls, strategic interdependence
complications emerge again.
No labor union will take
the lead
in accepting an extension of the indexation interval without the
guarantee that other labor unions will "do the same.
Summing
the implicit trigger point argument shows that inertia
is
up,
an
obstacle to inflation deceleration, but not to inflation
acceleration.
Turning back to our staggered wage setting model, let
us assume that for t
~
O
inflation runs at a constant
rate
n
with equilibrium wage indexation:
=
z
1T
l-e
-n
Q(T)
(T ~ O)
(26 )
~
{27}
(T
O)
At time t=O the government announces a stabilization
program eventually intended to achieve zero inflation and
employment.
full
This requires some spontaneous or managed
de-indexation scheme where alI peak/average ratios
wage contracts are eventually reduced to one.
in
Because
strategic interdependence and lack of synchronization
nominal
of
of
both
wage
adjustment dates, de-indexation cannot escape three complications.
First, in the absence of income policies,
the
government can, at best, determine the nominal output path, but
.25.
not the price path, which is entirely dependent on nominal wage
increases.
Why then, in the absence of unemployment, any labor
group should take the lead in accepting lower real wage peaks,
can only be explained by imprudent playing that only yields
perfect foresight if the other players are equally imprudente In
a word, staggered wage setting with high inflation rates
makes
a strong case in favor of income policies.
Second, even if an ideal policy-mix leads to
foresight, the price path for t
make
Z(T)
~
Z
~
O
unemployment.
path with
must be chosen so
for ali labor groups.
foresight, workers will only accept
perfect
as
In fact, under
Z(T)
<
Z
in
to
perfect
case
of
The neutrality dream, a declining inflation
Z(T)=Z
rate
for ali, is a mathematical impossibility
according to theorem 3.
Yet, as will be shown below,
foresight price paths that yield
transition phase do existo
Z(T)
>
Z
oerfect
during the
The problem is that they
whole
may
be
rather complicated, and that simpler paths may imbalance the real
wage structure.
Third, unless nominal wagescan be cut, gradualism is
inevitable, some residual price increase being necessary
reduce from peak to average the purchasing power of the
to
labor
groups adjusted just before the announcement of the stabilization
programo
Prices can only be stabilized at a levei Q(f) such that:
Q(f)
Q(O)
~
(28)
TI
l-e
-TI
As an example, if the inflation rate runs at 300%
year before the announcement of the stabilization plan
wages are adjusted every six months (which makes
~=ln
and
a
if
2=0,6931),
.26.
the residual price increase must be, at least:
Q(f)-Q(O) _ 38,63%
Q(O)
Even if nominal wages can be cut, a shock treatment
intendend to make Q(t)=Q(O) is an awkward policy choice. In fact,
according to equation (25), this would require:
l
ZQI (t) = W (t) = S(t)-S(t-l) = O
for
alI t ~ O
which, with stable prices, would perpetuate a disequiIibrium \vage
structure, where nominal wages of class
x(O
<x~
1) would
be
frozen at:
s (x) =
TI
e (x-I)
l-e
-TI
z Q(O)
standing below the equilibrium leveI zQ(O) for
and
above zQ(O) for higher values of x.
An attractive incomes-policy arrangement is the D-day
proposal of former Brazilian Minister of Finance Octavio Gouveia
de Bulhões.
Nominal wages would be adjusted at the contractual
dates for price increases before the announcement
stabi~ization
t
~
O.
the
plan, but not for the residual inflation following
that announcement.
alI
of
In our model, this would imply S(t)=S(O) for
Prices, according to equations (22) and (25),would
move up so that:
QI (t)
=
TI
l-e
(l-e TI
-TI
namely, along the curve:
(t-l)
Q (O)
(O ~ t
~
1)
.27.
Q(O)
l-e- TI
Q (t)
=
Q(t)
= --TI
TI
(O ~ t~ 1)
(t ~
Q(O)
1)
l-e
The fact that the inflation gradually declines from
to zero without requiring nominal wage cuts and
stabilize with Z(1)=Z
that
TI
orices
for all labor classes after t=l
only
displays part of the attractiveness of the D-day proposal. A still
more interesting conclusion is that in the transition
< 1,
namely, for -1 < 1
period,
the average purchasing power in every
individual wage cycle is greater than the equilibrium level z.
For contracts signed before the D-day and that mature after
the
beginning of the stabilization program, the conclusion is obvious.
Such contracts would yield an average purchasing power
inflation rate was kept equal to
8ince the inflation
TI.
declines after t=O, it follows that Z(1)
For O
<
>
1 for
z if the
rate
-1 < 1 ~ O.
1< 1, according to equation (23), and
since
S (1) =k (T ) Q (T ) =8 (O) :
Z(T)
=
S (O)
J:+l
1
da
Q(o)
Hence:
z· (1) = 8 (O)
8ince Q(1+1)
{ Q
>
(~+l)
Q(1) for
Z(1) is a decreasing function of
one concludes that Z(1)
1
Q(1)
>
1
for
}
0<1<
1
O
for
<
T
1, it follows
O
<
1.
< 1'~
that
1. 8ince z(l)=z,
The possibility of
-------------------------------------------~
.28 .
having
Z(T)
>
I
in alI individual wage cycles while, at
any
time, the average real wage of all workers stands at W(t)/Q(t)=z
should not be taken as a paradoxo
In fact, we are dealing with
overlapping wage cycles, and the fact that
>
Z (T)
< T<
1 for -1
does not imply that the purchasing power of all labor grou?s
greater than
z in the transi tion period
O o:;;;; t
o:;;;;
is
1.
Assuming perfect foresight, the D-day price path might
be understood as a rational expectations response to the
announcement of a policy that would gradually reduce the rate of
expansion of nominal output from T
to zero.
In fact,
once
strategic interdependence is taken into account, one can hardly
belive that markets would spontaneously produce any arrangement
such as the D-day wage adjustrnent rule.
It may be hard to
implement,even as an incornes-?olicy decision, since labor unions
are not expected to understand the rnathernatics of staggered wage
setting.
It may be politically much easier to darnpen inflation
through a fractional indexation arrangernent:
S(t)-S{t-l)
S{t-l)
=
a
Q (t) -Q (t-l)
(O
<
a
<
technically inferior to the D-day proposal, but that
understood without
1)
Q(t-l)
can
be
much intellectual sophistication.
The foregoing analysis describes how staggered
setting cornplicates anti-inflationary policies,
conclusions applying to 3taggered price setting.
problern is that, since adjustrnent
the
The
wage
same
central
dates are not synchronized,
instant wage-price stabilization becornes virtually irnpossible.
In fact, at any instant some wages and prices are too high,sorne
too low, equilibriurn being reached across time by continuous
1
.29.
rotation of relative positions. Of course, a tempting proposal
is to immediately realign alI
~utting
wage~
and prices to equilibrium,
some and increasing others, depending on how recent the
last adjustment was. Yet in most cases this will be found illegal,
since it violates outstanding contracts. Moreover, it would
involve nominal wage cuts, which politicians consider
a
nightmare. Gradualism then becomes inevitable, the D-Day proposal
indicating the quickest possible achievement.
Still, there is one escape route for a shock treatment,
that of monetary reformo In fact, a currency change
is
not
necessarily limited to a cut of a certain number of zeros in the
country's monetary unit. It may well include a number of rules
that establish how contracts should be translated from the old
currency into the new one. Recent experiences such as the Austral
Plan of Argentina and the Brazilian Cruzado Plan indicate how
an imaginative use of such translation rules can lead
immediate relative wage and price
to
syn~hronization.
4) The Role of Monetary Reforms
Attempts to stop a big inflation are often coupled
with monetary reforms for two well known reasons. First, as a
political move to signal a new
age of price stability. Second,
as a practical action to increase the purchasing power of the
currency unit, which is usually achieved by cutting three, six
or even twelve zeros in the
old money bills. Needless to say,
changing names and cutting zeros is a fruitless exerci se unless
fiscal austerity is imposeq and the inflationary
expectations
abated. As in the case of income policies, the list of failed
·30.
rnonetary reforms is too 10ng to be ignored.
How a
~onetary
reform can erase the inf1ationary
psych010gy is an intriguing question, since it involves
contradictory p01icy answer.
a
Incomes policies may be essential
to break inertia, but economic agents must also be convinced
that the Central Bank will stop fueling infla tion by
printing.
money
Yet once the expected rate of inf1ation falls
dramatically, fiat money must be generously printed
for
period of time to rneet the increase in money demando
a
The
classic rule of thurnb adopted by the Gerrnan stabilization
program of November, 1923, and borrowed by the Argentine
Austral Plan, i.e., prohibiting money creation to finance public
sector deficits, has a strong appeal, since it suggests that
the rnoney supply will be properly kept under control
long
run.
in
the
How to bridge short terrn prob1em however, when
renewed confidence in the dornestic currency (based
on
the
assurnption that it will be no longer generously printed) requires
a lot of additional printing, rernains a complicated policy
issue.
Policy-makers rnust walk on a sharp knife-edge, avoiding
so quick a reliquiÍication that it rnight re-ignite inf1ationary
expectations,as
well as one so slow that it would
lead
to
recession caused by a credit crunch.
The best solution might well be to set targets
for
total financial assets held by the public (U 4 ), allowing Ml to
expand as long as other financial assets decline.
Yet, since
M4 statistics do not inclucbd foreign currency held bythe public,
the rule rnust be arnended in case th2re i5 a significant shift
from dollar assets to domestic currency securities.
Moreover,
when the inflation rate falls from 400% or 2,000% a year
to
. I
.31.
virtually zero, there is no guarantee that the
demand for
financiaI assets will be kept unchanged in real terms.
Hence,
besides tracking M4 as a basic guide-line the Central Bank
should keep a close eye on what is actuaily happening to nominal
GNP.
Monetary reforms can
al~0
~erform
a veryimportant
function, that of dissipating the Fischer-Taylor inertia.
As
previously discusseq, a serious obstacle to overnight price
stabilization is that outstanding non-indexed contracts reflect
old inflationary expectations.
Honoring such contracts when
the inflation rates falI dramatically implies unanticioated
wealth transfers between contracting parties that may lead
bankruptcies and recession.
Of course, high inflation
rates
limit non-indexed contracts to short maturities, but even
the transfer effect may be sizeable.
Let us assume,
to
50
for
instance, that the inflation rate falls abruptly from 15%
a
month to zero, as occurred in Brazil on February 28,1986.
In
the absence of a rnonetary reforrn, a prornissory note rnaturing
three rnonths after the stabilization day would cost
borrower a 52% unanticipated real surcharge.
the
As to staggered
wage-setting, the obstacles it crates to immediate
stabilization were extensively discussed in the
price
preceding
section.
Monetary reforms can solve alI these probldS created
by unanticipated stabilization and by lack of synchronization
by establishing appropriate translation rules.
In fact, once
a new currency is substituted for the old one, the law
must
state how outstanding contracts should be rewritten in the new
monetary unit.
The imagina tive innovation of the
Argentine
.32.
Austral Plan, borrowed by the Brazilian Cruzado Plan, was to
recognize that there is no legal or economic reason why
the
translation rule for currency bills should hold for other
~obligations
and contracts. In the case of non-indexed liabilities,
the natural way to eliminate the effects of unanticipated
stabilization is to establish conversion rates declining by a
daily factor corresponding to the old inflation rate. As
example, the Cruzado reform immediately
an
converted cruzeiro bills
and demand deposits into cruzados by a cut of three zeros.
Conversion rates for cruzeiro-denominated liabilities with
future maturities were set to decline by a daily factor, starting
February 28
th
, of 1.0045. Thus, a promissory note of one million
cruzeiros maturing April 30
th
1986 was worth 770.73 cruzados,
not 1.000 cruzados. This specific translation rule was borrowed
from Argentine Austral Plano
Neither Argentina nor Israel had to f&ce problems of
staggered wage setting, since in both cases wage adjustment dates
were already synchronized. Brazil did, and the true innovation
of the Cruzado Plan was to convert wages from cruzeiros
into
cruzados by computing their average purchasing power of
the
last six months with an increase of 8%. This actually meant
reducing the nominal wages of the labor groups adjusted
in
January and February, 1986, and in all cases increasing wages
substantially below the past inflation rate. The 8% bonus was
intended to offset the inflationary effect of a number of price
increases which were to be enforced
o~ Feb~uary
28
th
1986,
50
that the plan was to synchronize alI real wages at their average
leveI of the previous six months, since the lengfi~ of nominal
wage contracts was exactly six months. At the last moment
the
.33.
government decided not to enforce these sectoral price
increases,
50
that average real wages were actually increased
by 8%. This may have been the original sin of the Cruzado Plano
5 - Incomes POlicy Matching
The previous discussion on strategic interdependence
among price-setters provides the rationale for combining monetary
reforms with income policies and fiscal austerity. How incomes
policies should be designed is a challenging issue. While a low
calory menu seems enough to break inertia after a true hyperinflation,
when prices expand by factors of millions or billions a year,
much more sophisticated cooking is required when the problem is
to stop a three or four digit inflation.
In fact, in a true hyperinflation (where, according
to Cagan's definition, prices continuously increase above 50% a
month) money not only loses its function as a store of value,
but also ceases to perform its role as an unit of account, wages
and prices being usually set in some foreign currency. In this
case, exchange-rate stabilization serves as a universal incomes
policy tool. The classic success story is the German stabilization
plan of November, 1923.
Complications arise when infla"tion is
50
high as
to
trigger widespread backward-looking indexation arrangements
(since instantaneous price-index links never proved feasible),
but not to the point of quoting wages and prices in foreign
currencies. The unit of account, in such cases, is neither the
domestic currency nor the exchange rate, but some lagged price
indexo Here irnbalanced income policies may lead to a disaster,
----------------------------------------
.34.
even if matched by fiscal austerity, as happened with Chile in
1979.
The exchange rate was pegged at 39 pesos per dollar,the
fiscal deficit was eliminated (actually it was turned
into
a
surplus), but wages continued to be indexed for past inflation.
As a result, the only force to dampen inflationary inertia was
real exchange-rate overvaluation.
Inflation rates
fell from 40% in 1978 to 9% in 1981.
actually
Yet exchange-rate
overvaluation led to dismal recession and to a wasteful increase
in foreign debt.
Summing up, income policies were seriously
mismatched; once the exchange rate was frozen, wages
should
also have been frozen, perhaps with the political help
of
a
temporary price freeze.
One may argue that the exchange rate has been fixed and
once nominal wages have been frozen, there is no need to freeze
prices, provided both aggregate demand and monopoly profit
margins are kept under controlo
This may be absolutely correct
from the technical point of view, but an income policy package
without price controls risks being a commodity with too
supply and no demand at alI.
much
In fact, the political marketing
of income policies is largely dependent on its visible content,
namely, on price freezes.
that it may be the
It should be recognized, however,
pDison~pill
of the stabilization package. In
fact, once prices are frozen, inflation falls to zero by
definition.
Yet there exists the risk that price inflation will
be replaced by queue inflation, not necessarily a better
achievement in terms of social welfare.
In a word, policy-makers and professional economists
should agree on some basic principIes of an anti-inflationary
code:
a)price stabilization cannot be sustained without
.35.
aggregate demand managementi
b) fiscal austerity is the
key
issue as far as aggregate demand is concerned, except in a few
countries with excess savingsi
c) inflationary inertia
true problem, since common knowledge can never be
is
a
easily
50
spread as assumed in the rational expectations macroeconomic
literaturei
d) in come policies, therefore, are an
instrument to fight big inflatíonsi
essential
e) there is no sense
discussing whether inflation should be tackled by the
in
supply
side or by the demand side, a concerted approach being
necessary, at least when the problem is to stop a big inflation;
f) stopping a big inflation involves the design of an equilateral
triangle, the vertices of which are fiscal equilibrium, income
policies and monetary reformi g) income policies can hardly be
imposed without price controls, the political counterpart of a
wage freeze being a general price freezei h) price freezes may
be the pOison-pill of a stabilization packgei i) wages,prices
and exchange rates should never frozen out of relative
price
equilibriumi j) the effectiveness of income policies can only
be tested after they are abolished.
.36.
APPENDIX
THE MATHEMATICS OF STAGGERED WAGE SETTING
We shall now prove the three basic theorems
on
the
mathematics of staggered wage setting mentioned in the preceding
texto
Key concepts and relations are provided by the following
equations:
W(t)
(A. 1)
S (T) dT
W(t)
zQ(t)
(A. 2)
(A. 3)
k (T)
=
S (T)
Q(T)
z (T)
=
S{T)
J'+1 Q~()")
,
do
(A.4 )
From (A.4) it follows that, if 01 (0)/0(0) = TI
constant, then Q(o)
Z(T)
= Q(t)
l-e- TI
= k(T)
e
TI (O-T)
is
a
, and:
(A. 5)
TI
Theorem 1:
for
t
~
(The constant inflation rate theorem): If,
O, Q{t)=Q{O)e TIt , TI indicating a positive constant,then:
lim z{t) = z
t-HX>
and:
.37.
lim
k (t)
-~.;....=
z (t)
t~oo
1T
----1T
l-e
, Proof:
Taking derivatives in both sides of (A. I):
=
W' (t)
=
zQ' (t)
5 (t) -5 (t-l)
Dividing by (A.2):
=
''lT
Q' (t)
Q (t)
O
<
e-TI
=
5(t)-S(t-l)
zQ (t)
= __l__ {
k(t)
z
k(t-l) =TI,Z
k(t) -
since
W' (t)
W (t)
=
<
1, any solution of this difference equation
converges to:
lim
k(t)
= -z.'If
--1T
l-e
T~OO
It follows from (A.5) that:
lim
z(t)
=
z
t~oo
and, as a consequence:
lim
t~oo
k (t)
z (t)
=
_1T-,--_
l-e
-TI
Before dealing with the peakjaverage ratio theorem, let
us prove the:
.38.
Lemma 1: rf
-<
Ir I
1, any solution of the differential-
diference equation:
(A. 6)
x' (t) = r(X(t) - X(t-l) )
converges to a constant x.
Proof: For every integer n, let us define:
~n = max{1 X' (t) I
I X'
rf
(T
)1
= v
n
n-l
~
t
~
n}
the
,then (A.6), combined with
Lagrange theorem, implies:
V
Ix'
(T-e)
~
n
I~
=Ix'
(T)
I
=
Ir X'
(T-e)1
where O <
e < 1.
max {v n ' v - l } and since Irl < 1, thisimplies
n
Irl max {v ,v l} , or equivalently:
n nv n-l
(A. 7)
It follows that, lim Ix' (t)! ~ lim
t+ oo
n+ oo
According to (A. 7), for any O ~ s
00
00
IX' (s+n) I
L
Since
~
L
n=l
n=l
This implies that the series:
00
f(s)
=
L
n=l
X'(s+n)
<
1
1
l-!r\
.39.
converges, since it is absolutely convergente
From (A.6) it follows that:
lim
n
X(s+n)
=
f(s)
X(s) +
r
+00
To complete the proof of the lemma it remains
prove that if O.ç s <: s' ..:::: 1, then X(s') +
+
f (s ')
r
=
to
X (s)
+
f (s) , or equi valently that:
r
lim
X(s'+n) - X(s+n)
=
O
Using once again Lagrange's theorem:
X(s'+n) - X(s+n) = (s'-s)X' (a+n)
(s< a
< s')
which yields:
lim
IX(s'+n)-X(s+n) I
~
lim
h-+ oo
n+ oo
vn
=
O
Theorem 2 (The peak/average ratio theorem): lf, for alI
t
~
O, k(t)
= rz,
r indicating a constant greater than one, then
the inflation rate Q' (t)/Q(t) converges to
r
=
TI
1-
e-TI
r, where:
·•4.0 •
Proof:
~
S(T)= rzQ(T), for T
for
t
~
O. Hence, by equations (A.l) and (A.2),
1:
Q(t)
~t:l
=r
(A. 8)
Q(t)dt
Making:
X{t)= Q(t)e- nt
(A. 9)
(A.8) is equivalent to:
X(t) = r
J
en{T-t) X(T)dT
t
(A .10)
t-l
From (A.9):
Q I (t)
=
Q(t)
Hence, to prove the theorem, it suffices to prove that
X(t) converges to a positive constant.
Taking derivatives in both sides of (A. la):
Xl (t) = re -1T (X (t) -X (t-l)) =
Since re- n = ___Tf_..__
eTf -1
>
=
O.
min { X (t)
I
(t-l) )
(A. 11)
To complete the proof, we just
For any integer n let us define:
m (n)
(X (t) -X
< 1, it follows from lemma 1 that
X{t) converges to a constant X.
need to show that X
1T
n-l .. t
.. n }
.41.
Since the price index Q(t) is always positive, m(n»O.
, where n-l
Let us assume that X(t) = m(n)
~
t
~
If
n.
t=n, equation (A. lO) yields:
j
TI (T-n)
dT=
n en (T-n) X(T)dT ;. m(n)r (n e
n-l
)n-l
X(n)=m(n)=r
the equality requiring X(T)=X(n)=m(n) for alI n-l
case,
(A.II) yields X(t)=m(n)
>
O for alI
If X(t)=m(n) and n-l
<
t
~
t
~
~T<n.
I!l
(n)
Inthis
n-l.
n, equation (A.IO) leads to:
J
n-l
~
X(t)= m(n)
rm (n-l)
t-l
which implies:
m(n)
~
m(n-l)
In either case, m(n) is a non-decreasing sequence
positive real numbers, which ryroves that X
Theorem 3:
>
O.
(The non-neutrality theorem).
If the
inflation rate decreases for t-l ~ T ~ t+l, then Z(T)
some t-l
~ T~
of
>
Z for
t.
Proof: We shall prove that the assumption of Z(T)~ z,
for alI
t-l
~ T~
t, is inconsistent with that of a declining
inflation rate.
In fact, to say that the inflation rate decreases for
t-l
~ T ~
t+l, is equivalent to saying that, in that interval,
the function:
q(a)
=
In Q{a)
.42.
1s strictly concave.
Hence, if m=q' (t), the curve lies below
the tangent at the point (t;q(t», namely:
m(a-t)
q(a)-q(t) : >; ;
the equality holding only when a=t.
Q(a)
: >; ;
It follows that:
Q(t)em(a-t)
and as a consequence that, for t-l : >; ; T : >; ; t:
I
Q(a)
da
>
I
Q (t)
Let us assume that
Z
l-e -m
m
(T) : >; ;
Z
e
m(t-T)
for alI t-l : >; ; T::>;;; t.
It
follows, from (A.2) and (A.4) that:
S (T)
< ___m__
-m
l-e
w(t)em{T-t)
(A .12)
Introducing (A.12) in (A.I) yields the contradiction
W{t) <W(t).
REFERENCES
1- ARIDA, P. and A.Lara Resende -(1985)- rnertia1 Inf1ation and
Monetary Reform in Brazi1: in John
Inf1ation and Indexation.
~!i11iamson
(ed.)
Institute for Internationa1
Economies.
2- DORNBUSCH, R. and M.H.Simonsen -(1986)- Inf1ation Stabi1ization
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e
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FGV
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ENSAIOS ECONOMICOS DA EPGE
(a partir de
n~
50)
50. JOGOS DE INFORMAÇÃO INCOMPLETA: UMA INTRODUÇÃO - Sérgio Ribeiro da Costa
Werlang - 1984 (esgotado)
51. A TEORIA MONETARIA MODERNA E O EQUILrBRIO GERAL WALRASIANO COM UM NOMERO
INFINITO DE BENS - A. Araujo - 1984
52. A INDETERMINAÇÃO DE MORGENSTERN - Antonio Maria da Silveira - 1984
53. O PROBLEMA DE CREDIBILIDADE EM POLrTICA ECONOMICA - Rubens Penha Cysne 1984 (esgotado)
54.
U~A ANALISE ESTATrSTICA DAS CAUSAS DA EMISSÃO DO CHEQUE SEM FUNDOS: FORMULAÇÃO DE UM PROJETO PILOTO - Fernando de Holanda Barbosa, Clovis de Faro e
Aloísio Pessoa de Araujo - 1984
55. POLrTICA MACROECONOMICA NO BRASIL: 1964-66
~
Rubens Penha Cysne - 1985 -
(esgotado)
56. EVOLUÇÃO DOS PLANOS BÁSICOS DE FINANCIAMENTO PARA AQUISiÇÃO DE CASA PROPRIA
DO BANCO NACIONAL DE HABITAÇÃO: 1964-1984 - Clovis de Faro - 1985 (esgotado)
57. MOEDA INDEXADA - Rubens P. Cysne - 1985 (esgotado)
58. INFLAÇÃO E SALARIO REAL:
1985 (esgotado)
A EXPERIENCIA BRASILEIRA - Raul José Ekerman -
59. O ENFOQUE MONETARIO DO BALANÇO DE PAGAMENTOS: UM RETROSPECTO - Valdir
Ramalho de Melo - 1985 (esgotado)
60. MOEDA E PREÇOS RELATIVOS: EVIDENCIA EMPrRICA - Antonio SaLazar P. Brandão 1985 (esgotado)
61. INTERPRETAÇÃO ECONOMICA, INFLAÇÃO E INDEXAÇÃO - Antonio Maria da Silveira 1985 (esgotado)
62. MACROECONOMIA - CAPrTULO I - O SISTEMA MONETARIO - Mario Henrique Simonsen
e Rubens Penha Cysne - 1985 (esgotado)
63. MACROECONOMIA - CAPrTULO I I - O BALANÇO DE PAGAMENTOS - Mario Henrique
Simonsen e Rubens Penha Cysne - 1985 (esgotado)
64. MACROECONOMIA - CAPrTULO I I I - AS CONTAS NACIONAIS - Mario Henrique Simonsen
e Rubens Penha Cysne - 1985 (esgotado)
65. A DEMANDA POR DI VI DENDOS: UMA JUSTI FI CATI VA TEORI CA - TOMMY CH I N-CH I UTAN e
Sérgio Ribeiro da Costa Werlang - 1985 (esgotado)
66. BREVE RETROSPECTO DA ECONOMIA BRASILEIRA ENTRE 1979 e 1984 - Rubens Penha
Cysne - 1985 (esgotado)
67. CONTRATOS SALARIAIS JUSTAPOSTOS E POLfTICA ANTI-INFLACIONARIA - Mario
Henrique Simonsen - 1985
68. INFLAÇÃO E POLrTICAS DE RENDAS - Fernando de Holanda Barbosa e Clovis de
Faro - 1985 (esgotado)
69. BRAZIL INTERNATIONAL TRADE ANO ECONOMIC GROWTH - Mario Henrique
Simonsen - 1986
70. CAPITALIZAÇÃO CONTrNUA: APLICAÇÕES - Clovis de Faro - 1986 (esgotado)
71. A RATIONAL EXPECTATIONS PARADOX - Mario Henrique Simonsen - 1986 (esgotado)
72. A BUSINESS CYCLE STUDY FOR THE U.S. FORM 1889 TO 1982 - Carlos Ivan
Simonsen Leal - 1986
73. DINAMICA MACROECONOMICA - EXERCrCIOS RESOLVIDOS E PROPOSTOS - Rubens Penha
Cysne - 1986 (esgotado)
74. COMMON KNOWLEDGE ANO GAME THEQRY - Sérgio Ribeiro da Costa Werlang - 1986
75. HYPERSTABILITY OF NASH EQUILIBRIA - Carlos Ivan Simonsen Leal - 1986
76. THE BROWN-VON NEUMANN OIFFERENTIAL EQUATION FOR BIMATRIX GAMES Carlos Ivan Simonsen Leal - 1986 (esgotado)
77. EXISTENCE OF A SOLUTION TO THE PRINCIPAL'S
Leal - 1986
PROBLEM - Carlos Ivan Simonsen
78. FILOSOFIA E POLTTICA ECONOMICA I: Variações sobre o Fenômeno, a Ciência e
seus Cientistas - Antonio Maria da Silvei ra - 1986 (esgotado)
79. O PREÇO DA TERRA NO BRASIL: VERIFICAÇÃO DE ALGUMAS HIPOTESES - Antonio
Sal azar Pessoa Brandão - 1986
80. MtTODOS MATEMATICOS DE ESTATfsTICA E ECONOMETRIA: Capitulos 1 e 2
Carlos Ivan Simonsen Leal - 1986
81. BRAZILIAN INDEXING ANO INERTIAL INFLATION: EVIDENCE FROM TIME-VARYING
ESTIMATES OF AN INFLATION TRANSFER FUNCTION
Fernando de Holanda Barbosa e PaulO. McNelis - 1986
82. CONSORCIO VERSUS CRtDITO DIRETO EM UM REGIME DE MOEDA ESTAvEL-~ Clovis de Faro
:'1986
83. NOTAS DE AULAS DE TEORIA ECONOMICA AVANÇADA I - Carlos Ivan SimonsenLeal-1986
84. FILOSOFIA E POLTTICA ECONOMICA I I - Inflação e Indexação - Antonio' Maria da
Silveira - 1986
85. SIGNALLING AND ARBITRAGE - Vicente Madrigal
e
Tommy C. Tan - 1986
86. ASSESSORIA ECONOMICA PARA A ESTRATtGIA DE GOVERNOS ESTADUAIS: ELABORAÇÕES
SOBRE UMA ESTRUTURA ABERTA - Antonio Maria da Silveira - 1986
87.
THE CONSISTENCY OF WELFARE JUDGEMENTS WITH A REPRESENTATIVE
CONSUMER - James Dow e Sérgio Ribeiro da CostaWerlang
88. INDEXAÇÃO E ATIVIDADE AGRfCOLA: CONSTRUÇÃO E JUSTIFICATIVA PARA A ADOÇÃO DE
UM fNOICE ESPECrFICO - Antonio Salazar P. Brandão e Clóvis de Faro - 1986
89. MACROECONOMIA COM RACIONAMENTO UM MODELO SIMPLIFICADO PARA ECONOMIA ABERTA
- Rubens Penha Cysue, Carlos Ivan Simonsen Leal e Sergio Ribeiro da Costa
Wer1ang - 1986
90. RATIONAL EXPECTATIONS, INCOME POLICIES ANO GAME THEORY - Mario Henrique
Simonsen - 1986
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