Hyperinflation in Brazil, Israel, and Nicaragua revisited

0.25 MB·

by  M. A. Szybisz

arXiv-hyp-BIN-v3.tex
Hyperinflation in Brazil, Israel, and Nicaragua revisited
Martin A. Szybisz
Departamento de Economía, Facultad de Ciencias Económicas, Universidad de Buenos Aires,
Av. Córdoba 2122, RA–1120 Buenos Aires, Argentina
∗
Leszek Szybisz
Laboratorio TANDAR, Departamento de Física, Comisión Nacional de Energía Atómica,
Av. del Libertador 8250, RA–1429 Buenos Aires, Argentina
Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, RA–1428 Buenos Aires, Argentina and
Consejo Nacional de Investigaciones Científicas y Técnicas,
6
Av. Rivadavia 1917, RA–1033 Buenos Aires, Argentina
1
(Dated: April13, 2016)
0
2
Theaimofthisworkistoaddressthedescriptionofhyperinflationregimesineconomy. Thespirals

r ofhyperinflationdevelopedinBrazil, Israel, andNicaragua arerevisited. Thisnewanalysisofdata
p indicatesthattheepisodesoccurredinBrazilandNicaraguacanbeunderstoodwithintheframeof
A
themodelavailableintheliterature,whichisbasedonanonlinearfeedback(NLF)characterizedby
an exponent β >0. In the NLF model the accumulated consumer price index carries a finite time
2 singularity of the type1/(t −t)(1−β)/β determining a critical time t at which theeconomy would
1 c c
crash. ItisshownthatinthecaseofBraziltheentireepisodecannotbedescribedwithauniqueset

of parameters because the time series was strongly affected by a change of policy. This fact gives
]
N support to the “so called” Lucas critique, who stated that model’s parameters usually change once
policy changes. On the other hand, such a model is not able to provide any t in the case of the
G c
weaker hyperinflation occurred in Israel. It is shown that in this case the fit of data yields β →0.
. This limit leads tothelinear feedback formulation which does not predict any t . Anextension for
n c
i theNLFmodel is suggested.
f
-
q PACSnumbers: 02.40.XxSingularitytheory; 64.60.F-Criticalexponents; 89.20.-aInterdisciplinaryapplica-
[ tionsofphysics;89.65.GhEconophysics;89.65.-sSocialsystems

3
v There are 1011 stars in the galaxy. That used to be a finite time singularityofthe form1/(t −t)(1−β)/β. This
c
2 huge number. But it’s only a hundred billion. It’s less feature allows to determine a critical time t at which
c
9
than the national deficit! We used to call them astro- theeconomywouldcrash. Althoughthismodelhasbeen
0
nomical numbers. Now we should call them economical successfully applied to many cases [4–8], the authors of
0
0 numbers. Ref. [4]found difficulties indetermining tc for regimesof
. Richard Feynman hyperinflation occurred in Brazil, Israel, and Nicaragua.
1
Therefore, the present work is devoted to revisit these
0
episodes. It is shown that after a revision of data is
6
1 I. INTRODUCTION possible to predict reasonable values of tc for Brazil and
: Nicaragua. However, in the case of Israel the difficulty
v
persists, this feature would be plausibly attributed to
i
X Inflationcontributionisfundamentaltoreachthe“eco- permanent but partial efforts for stopping inflation. In
r nomical numbers” quoted by Feynman (see data plotted order to follow better the evolution of inflation we pro-
a below). But most importantly, when inflation surpasses vide brief historical descriptions for these countries.
moderate levels it affects real economic activities. Mod-
The paper is organized in the following way. In Sec.
els of hyperinflation are especially suitable to emphasize II the NLF theory is outlined with details in order to
that inflation implies bad “states of nature” in economy.
present self-consistently the tools applied for analyzing
Wars,statesbankruptcies,andchangesofsocialregimens regimes of hyperinflation. The episodes occurred in
arethecharacteristicsofsuchregimens. Theseissuesare
Brazil, Israel, and Nicaragua are revisited in Sec. III.
analyzed in textbooks on econophysics [1–3].
Finally, Sec. IV is devoted to summarize conclusions.
Themodelforhyperinflationavailableintheliterature
is basedon a nonlinear feedback (NLF) characterizedby
an exponent β > 0 of a power law. In such an approach
II. THEORETICAL BACKGROUND
the accumulated consumer price index (CPI) exhibits a
Letus recallthatthe rateofinflationi(t) isdefined as
∗Corresponding author; Electronic address: i(t)= P(t)−P(t−∆t) = P(t) −1, (2.1)
[email protected] P(t−∆t) P(t−∆t)
2
∗
whereP(t)istheaccumulatedCPIattimetand∆tisthe the fact that the expected inflation P (t) expands the
period of the measurements. In the academic financial response to the price level P(t) in order to forecast and
literature, the simplest and most robust way to account meet the inflation of the next period. Now, one may
for inflation is to take logarithm. Hence, the continuous introduce the expected GRI
rate of change in prices is defined as
∗
∆t ∆t P (t+∆t)
∗ ∗
C(t)= ∂lnP(t) . (2.2) r (t+ 2 )≡C (t+ 2 )∆t=ln P∗(t) . (2.9)
(cid:20) (cid:21)
∂t
So, expressions (2.7) and (2.8) are equivalent, respec-
Usually the derivative of Eq. (2.2) is expressed in a dis-
tively, to
crete way as
∆t ∆t
∆t [lnP(t+∆t)−lnP(t)] r(t+ )=r∗(t− ), (2.10)
C(t+ ) = 2 2
2 ∆t
1 P(t+∆t) and
= ln . (2.3)
∆t P(t)
(cid:20) (cid:21) ∆t ∆t
∗
r (t+ )=r(t− ). (2.11)
The growth rate index (GRI) over one period is defined 2 2
as
These relations imply
∆t ∆t P(t+∆t)
r(t+ ) ≡ C(t+ )∆t=ln
2 2 P(t) r(t+∆t)=r(t−∆t), (2.12)
(cid:20) (cid:21)
= ln[1+i(t+∆t)]=p(t+∆t)−p(t),
giving a constant finite GRI equal to its initial value
(2.4)
r(t)=r(t )=r . The accumulatedCPI evaluated using
0 0
Eq. (2.6) leads to an exponential law
where a widely utilized notation
t−t
p(t)=lnP(t), (2.5) P(t)=P exp r 0 , (2.13)
0 0
∆t
(cid:20) (cid:18) (cid:19)(cid:21)
was introduced. It is straightforward to show that the
accumulated CPI is given by where P0 =P(t0).
1 t
′ ′
P(t)=P(t ) exp r(t)dt . (2.6)
0
∆t B. Feedback contribution to the equation for
(cid:20) Zt0 (cid:21)
inflation
A. Cagan’s model of inflation
DuetothefactthattheCPI duringspiralsofhyperin-
flationgrowsmorerapidlythantheexponentiallawgiven
Inhispioneeringwork,Caganhasproposed[9]amodel by Eq. (2.13), the Cagan’s model for inflation has been
of inflation based on the mechanism of “adaptive infla- generalized by Mizuno, Takayasu, and Takayasu (MTT)
tionary expectation” with positive feedback between re- [10] including a linear feedback (LF) process. For this
alized growth of the market price P(t) and the growth purpose, the relation (2.7) was kept, while Eq. (2.8) was
∗
of people’s averaged expectation price P (t). These two replaced by
prices are thought to evolve due to a positive feedback
mechanism: an upward change of market price P(t) in P∗(t+∆t) P(t)
ln = (1+2a ) ln ,
a unit time ∆t induces a rise in the people’s expecta- P∗(t) p P(t−∆t)
∗ (cid:20) (cid:21) (cid:20) (cid:21)
tion price P (t), and such an anticipation pushes on the
(2.14)
marketprice. Cagan’s assumptions may be cast into the
following equations:
which leads to
∗ ∗
P(t+∆t) P (t) P (t)
1+i(t+∆t)= = = , (2.7) r(t+∆t)=r(t−∆t)+2a r(t−∆t). (2.15)
P(t) P(t) P∗(t−∆t) p
Here a is a positive dimensionless feedback’s strength,
and p
in fact, MTT defined a parameter B = 1+2a . In
MTT p
∗
P (t+∆t) P(t) the continuous limit one arrives at
= =1+i(t). (2.8)
P∗(t) P(t−∆t)
dr a t−t
= p r(t) → r(t)=r exp a 0 .
Actually P∗(t)/P(t) indicates that the process induces dt ∆t 0 p ∆t
(cid:20) (cid:18) (cid:19)(cid:21)
a non exact proportional response of adaptation due to (2.16)
3
In this approachthe CPI growsas a function of t follow- For 0<β <1 one gets
ing a double-exponential law [6, 10], so one gets
1−β
r β t −t t −t β
p(t)=p + 0 c 0 c 0 −1 .
lnP(t)=p(t)=p0+ r0 exp ap t−t0 −1 . 0 1−β (cid:18) ∆t (cid:19)(cid:20)(cid:18) tc−t (cid:19) (cid:21)
a ∆t (2.24)
p (cid:26) (cid:20) (cid:18) (cid:19)(cid:21) (cid:27)
(2.17) Thissolutioncorrespondstoagenuinedivergenceofp(t),
Since in practice there are cases where P(t) grows thelog-CPIexhibitsafinite-time singularityatthe same
more rapidly than a double-exponential law, in a next criticalvaluet asGRI.Letusemphasizethatallthefree
c
step, Sornette, Takayasu,and Zhou (STZ) [4] included a parametershavetheir ownmeaning: t isthe hyperinfla-
c
nonlinear feedback process in the formalism. In this ap- tion’send-pointtime;β istheexponentofthepowerlaw;
proach,Eq.(2.7)isalsokept,whilstEq.(2.14)isreplaced r is the initial slope for the growth of log-CPI; and p
0 0
by is the initial log-CPI. Equation (2.24) has been used for
the analysis of hyperinflation episodes reportedin previ-
∗
P (t+∆t) P(t) ous papers [4–8].
ln = ln
P∗(t) P(t−∆t)
(cid:20) (cid:21) (cid:20) (cid:21)
β
P(t)
× 1+2a ln , III. HYPERINFLATION IN BRAZIL, ISRAEL
p(cid:26) (cid:20)P(t−∆t)(cid:21)(cid:27) ! AND NICARAGUA REVISITED
(2.18)
We shallnow revisit the episodes of hyperinflationde-
leading to veloped in Brazil, Israel, and Nicaragua performing a
studywithintheframeworkoftheNLFmodeloutlinedin
r(t+∆t)=r(t−∆t)+2a [r(t−∆t)]1+β . (2.19) theprevioussection. Thesecaseshavebeenalreadystud-
p
ied by Takayasu and collaborators [4, 10]. In particular,
Here β > 0 is the exponent of the power law. In in Ref. [4] the authors stated: “a fit of the price time se-
the discrete version of this NLF model, r(t) follows a rieswithexpression(15)givesanexponentαlargerthan
double-exponential law; while P(t) increases as a triple- 15andcriticaltimest intherange2020-2080,whichare
c
exponential law [5, 6]. Notice that for β = 0 this for- un-realistic” (sic). Equation(15)ofRef. [4]is equivalent
mulationretrievesthe LFproposalofMTT givenby Eq. to Eq. (2.24) of the present work and the parameter α
(2.15). written in terms of the exponent β is
Taking the continuous limit in Eq. (2.19) one obtains
the following equation for the time evolution of r α=(1−β)/β . (3.1)
Hence,theresultsofRef.[4]correspondtoβsmallerthan
dr a
= p [r(t)]1+β . (2.20) 0.07. In addition, the authors of Ref. [4] said that the
dt ∆t
results are not improved by reducing the time intervals
For β > 0 the solution for GRI follows a power law ex- over which the fits are performed. In seeking for how
hibiting a singularity at finite-time t [4–6] to overcome this problem, they found that reasonable
c
critical t are obtained after a simple change of variable
c
1 1/β t −t 1/β fromlnP(t)toP(t),i.e. byfittingP(t)insteadoflnP(t)
c 0
r(t)=r0(cid:20)1−βapr0β t−∆tt0 (cid:21) =r0 (cid:18) tc−t (cid:19) . with Eq. (2.24).
(2.21)
(cid:0) (cid:1)
The critical time tc being determined by the initial GRI A. Hyperinflation of Brazil and Nicaragua
r(t=t )=r , the exponent β, and the strength param-
0 0
eter a
p We shall now proceed to discuss the entire regimes
of hyperinflation occurred in Brazil (1969-1994) and
t −t 1
c 0
= . (2.22) Nicaragua(1969-1991). Insearchingwhyitwasimpossi-
∆t βa rβ
p 0 ble to describe satisfactorily well these episodes utilizing
the NLF model the data of both these countries were
In turn, the log-CPI for β 6=1 is obtained by integrating
revised.
r(t) according to Eq. (2.6)
LetusnowpresentashortstoryofBrazilianeconomic
P(t) t dt′ difficulties. In fact, Brazil was not defeated in a war nor
ln =p(t)−p = r(t′) wasrequiredtopaywarreparations,buttheforeigndebt
0
P ∆t
(cid:20) 0 (cid:21) Zt0 accumulatedinthe1970’sbyborrowinglargeamountsof
r1−β 1 1−ββ cheappetrodollars,theexternalshockof1979(secondoil
= 0 −1 . shockandinterestrateshock)andthesuspensionofnew
(1−β)ap(cid:26)(cid:20)1−βapr0β t−∆tt0 (cid:21) (cid:27) externalfinancingsince1982hadtogetherproducedsim-
(2.23) ilarconsequences[11,12]. Thecountrythatinthe1970’s
(cid:0) (cid:1)
4
1016 6
STZ*
Brazil
Brazil STZ* 5
1012 (a) NLF (b) NLF
4
) LF
P(t) 108 r(t 3
2
104 tc
t
c 1
100 0
1965 1970 1975 1980 1985 1990 1995 2000 1965 1970 1975 1980 1985 1990 1995 2000
t t
FIG.1: (a)SquaresareyearlyCPIinBrazilfrom1969to1999,normalizedtoP(t =1969)=1,presentedinasemi-logarithmic
0
plot. (b)CirclesareyearlyGRIforthesameperiodasin(a). Thesolid curvein(a)isthefitoflnP(t)from1969to1990with
Eq. (2.24), i.e. NLF model, while the dashed line is the fit of P(t) from 1969 to 1994 with Eq. (2.24) reported by STZ* (see
text). In panel (b) the solid curve NLF stands for r(t) evaluated with Eq. (2.21), while the solid curve LF is r(t) evaluated
with Eq. (2.16), in both cases theparameters quoted in Table I were used. The dashed curveis r(t) evaluated using Eq. (3.7)
and the dot-dashed curve is the asymptotic limit given by Eq. (3.8) (see text). In both panels the vertical solid line indicates
thecritical time t predicted bydata of theperiod 1969-1990.
c
received around 2% of gross domestic product (GDP) of aged the introduction of a new currency,put constraints
foreignsavings was now required to transfer resourcesof onpublicspendingandendedtheindexationoftheecon-
4to5%tothecreditorcountries. Debtservicewasequal omy. The new currency, the real, had a crawling peg
to 83% of export earnings in 1982. The country strug- againstthedollarasanominalanchorandwassomewhat
gledtofinanceitsexternalindebtednessandgrowthcame overvalued,whichmade importscheap,thus limiting the
to a halt. These economic problems were accompanied room for domestic producers to raise prices.
by political turbulence. The military dictatorship that Sornette, Takayasu, and Zhou [4] have analyzed the
had ruled Brazil since 1964 lost support and was forced complete series of CPI data from 1969 to 1994. The re-
to step down in 1985, which resulted in the return of sultsfromthefitofP(t),insteadofvaluesoflnP(t),with
democracy. the right-hand-side (r.h.s.) of expression (2.24) reported
SinceitsinaugurationinJanuary1985,thefirstdemo- by STZ in Table 2 of Ref. [4] are quoted in Table I and
cratic government after military rule exerted by the displayed in Fig. 1(a). For the sake of completeness, we
elected vice-president José Sarney (because the elected provide the relations between the parameters α, A, and
president Tancredo Neves fell ill) had limited means to B utilized by STZ and that used in the present work
resist spending pressure from congress. As a result, in-
flation, which had already been high for several years r0/∆t=αB/(tc−t0)1+α , (3.2)
thanks to the old practice of monetary financing of bud- β =1/(1+α), (3.3)
get deficits, frequent devaluations and indexation (au- p =A+B/(t −t )α . (3.4)
0 c 0
tomatic correction of prices, interest rates and wages
according to past inflation), ran totally out of control. We believe that the difficulty for fitting lnP(t) with
In 1987, the government was not able to pay the in- Eq.(2.24)arisesfromthefactthatin1991thereisanim-
terest on its foreign debt and Brazil’s public debt had portant departure from the initial trend clearly depicted
to be rescheduled. The inflation peaked at 2,950 % in in Fig. 1(b). The applied theory with a unique value of
1990. Thisbehaviorcanbe seeninFig.1(b), wheredata β is not able to describe the entire process. This feature
of yearly GRI computed using values taken from a Ta- is in agreement with the statement of Lucas [14] that
ble of the International Monetary Fund (IMF) [13] are parameters can change when economic policy changes.
displayed. A new elected president Fernando Collor de Therefore,inthepresentworkwefittedtoEq.(2.24)the
Melloappliedin1990theso-calledCollor’sPlaninorder dataoflnP(t)previousto1991only. Preliminaryresults
tostophyperinflation. AscanbeseeninFig.1(b)atthe have been already reported in Ref. [7]. The numerical
beginning the trend was changed, however, finally this task was accomplished by using a routine of the book
plan for stabilization failed [11, 12]. by Bevington [15] cited as the first reference in Chaps.
ThelaunchofthePlanoRealin1994wouldprovetobe 15.4 and 15.5 of the more recent Numerical Recipes [16].
the turning point. This plan, designedby Henrique Car- In practice, the applied procedure yields the uncertainty
doso, who would later become Brazil’s president, envis- in each parameter directly from the minimization algo-
5
TABLE I: Parameters obtained from theanalysis of episodes of hyperinflationoccurred in Brazil, Israel and Nicaragua.
Country Period Parameters Model χ
t a r β γ p
c p 0 0
Brazil 1969-1994 1997.50 0.402×10−2 0.058 1.93 STZa 0.604
1969-1990 1999.26±6.22 0.172 0.165±0.029 0.383±0.152 0 NLF 0.190
1990-1994 0.177±0.116 1.770±0.425 18.2 LF 0.158
Israel 1969-1985 1988.06 0.077 0.149 1.04 STZa 0.085
0.176±0.035 0.101±0.035 0 LF 0.088
2061±72 0.184 0.109±0.035 0.069±0.061 0 NLF 0.095
2527±456 0.177 0.102±0.035 0.010±0.009 0 NLF 0.089
1969-1984 0.178±0.045 0.100±0.040 0 LF 0.089
2048±79 0.189 0.107±0.041 0.080±0.093 0 NLF 0.094
2588±619 0.179 0.101±0.040 0.009±0.010 0 NLF 0.090
Nicaragua 1969-1991b 1992.91 0.881×10−5 0.063 3.24 STZa 0.848
1969-1987c 1987.71±0.87 0.383 0.101±0.031 0.710±0.217 0 NLF 0.298
1969-1988c 1992.32±2.38 0.316 0.067±0.020 0.356±0.102 0 NLF 0.519
a The valuesof theparameters listed in this line were calculated using those reported by STZ [4] obtained by fittingdata of
P(t) instead of lnP(t) to Eq. (2.24), see text.
b Datafrom Ref. [13].
c Data from Ref. [19].
rithm. In order to quantify the contribution of the feed- 1015
back we also evaluated a , which is given by [see Eq. STZ*
p LF
(2.22)] Brazil
1012
NLF
∆t
ap = βr0β(tc−t0) . (3.5) P(t) 109
The obtainedparameters,its uncertainties andthe root-
106
mean-square (r.m.s) residue of the fit, i.e. χ, are quoted
inTableI.Thedeterminedt isquitereasonableandthe
c
goodqualityofthisfitmaybeobservedinFig.1(a). The 103
GRI was calculated by using Eq. (2.21) and displayedin
Fig. 1(b), the theoretical results follows quite good the 1982 1986 1990 1994 1998
measured data. Vertical lines in both (a) and (b) panels t
indicate the obtained critical time t .
c
FIG. 2: Squares are yearly CPI in Brazil from 1982 to 1998,
Figure2clearlyshowsabifurcationbetweenthetrend
normalized to P(t = 1969) = 1, presented in a semi-
of data from 1969 to 1990 and that of data from 1990 0
logarithmic. The solid curve NLF is the fit of lnP(t) from
to 1994. Since there are only a few data points for the
1969to1990withEq.(2.24),whilethesolid curveLFstands
new incipient branch of hyperinflation, in order to have
for the fit of lnP(t) from 1990 to 1994 with Eq. (2.17). The
a quantitative description data of r(t) and lnP(t) from dashed line is the fit of P(t) reported bySTZ* (see text).
1990 to 1994 were simultaneously fitted with Eqs. (2.16)
and(2.17),respectively. Theobtainedparametersarein-
cludedinTableIandthefitsdenotedbyLFaredisplayed
Assuming that P(t) is given by Eq. (2.24) one gets
in Figs. 1(b) and 2.
Onthe other hand,one mayobserveinFig.2 thatthe 1/β
r tc−t0
fit reported by STZ* does not follow quite well the set r(t)= 0 tc−t . (3.7)
of measured CPI. The situation is even worse when one (cid:16) (cid:17) 1−β
examine the GRI. Accordingto the statementquoted on p0+ 1r−0ββ tc∆−tt0 ttcc−−tt0 β −1
the top of page 499 of Ref. [4] the accumulated P(t) is (cid:0) (cid:1)(cid:20)(cid:16) (cid:17) (cid:21)
theexponentialoftheintegralofr(t) asexpressedinEq.
Anevaluationofr(t)byusingthisformulawiththecorre-
(2.6) ofthe presentwork. The inverse,i.e. r(t), becomes
spondingparametersquotedinTableIyieldedthedashed
curve depicted in Fig. 1(b). The theoretical curve oscil-
dlnP(t) 1 dP(t) lates between both branches of measured data. It is in-
r(t)= = . (3.6)
d(t/∆t) P(t) d(t/∆t) teresting to notice that the asymptotic form of Eq. (3.7)
6
partner. The government spend a lot money to finance
TABLE II:Inflation in Nicaragua 1980-1997.
thewaragainsttheContrasduringthesecondpartofthe
Year Annualaveraged i(%) 80’s decade. The gap between decreasing revenues and
IMF-1a IMF-2b mushrooming military expenditures was filled by print-
1980 35.1 35.1 inglargeamountsofpapermoney. Inflationskyrocketed,
1981 23.8 23.8 peaking at 13,109%percent annually at the end of 1987.
1982 24.9 28.5 Inthelastfortyyears,atLatinAmericanlevel,thissitua-
1983 31.1 33.6
tionmaybeonlycomparedwiththatoccurredinBolivia
1984 35.4 141.3
in 1985, where the value 11,150% was reached. So it
1985 219.5 571.4
sounds like pretty much the same story as many other
1986 681.0 885.2
hyperinflation events: War debts and foreign pressures
1987 911.9 13109.5
inspire government to print large amounts of money. In
1988 14315.8 4775.2
1989 4709.3 7428.7 1988 began the efforts to stop the spiral of hyperinfla-
1990 3127.5 3004.1 tion, however, the success of this attempt was very poor
1991 7755.3 116.6 [17,18]. ItisworthwhiletonoticethatintheNicaragua’s
1992 40.5 21.9 administrationnoCentralBankexisted. IntheFebruary
1993 20.4 13.5 1990 Violeta Barrios de Chamorro won the elections de-
1994 7.7 3.7 feating Ortega. Afterwards, the usual stabilization pro-
1995 11.2 11.2
cedureswererigorouslyappliedleadingtoastableregime
1996 11.6 11.6
in 1992.
1997 9.2 9.2
Sornetteet al. [4]analyzedthe CPI from1969to1991
a Data from IMFWorld Economic Outlook (WEO) [13].
evaluatedwithdatatakenfromRef.[13],thecorrespond-
b Data from IMF (WEO) [19].
ing values of i(t) for the regime close to the hyperinfla-
tion’s peak are listed in Table II. Using the same proce-
dure as that applied in the case of Brazil these authors
for t→t becomes
c
fitted P(t) instead of lnP(t) with Eq. (2.24). The ob-
1−β ∆t tained results are included in Table I.
r (t)= , (3.8)
asympt β t −t The inflation occurred in Nicaragua during the 1980’s
(cid:18) c (cid:19)
decade is revisited in the present work. In so doing,
yielding a universal singularity (t −t)−1. This asymp- we found revised data of i(t) published by the IMF in
c
totic regime is reached quite soon as shown by the dot- the section of Economy in Ref. [19], which are also in-
dashed curve in Fig. 1(b). cluded in Table II. A glance at this table indicates that
Let us now refer to the hyperinflation occurred in the values at the beginning and at the end ofboth series
Nicaragua. During the decade of 1970’s this country are equal. However, the new values forming the peak
was involved in a civil war. In 1979, the Sandinista Na- of hyperinflation are shifted one year towards prior date
tionalLiberationFront(FSLN) overthrewAnastasioSo- and, in addition, the years corresponding to the values
moza Debayle, ending the Somoza dynasty, and estab- of about 7000%and 3000%are interchanged. The latter
lished a revolutionary government in Nicaragua. This featurechangestheprofileofthecrossovertothestation-
new government, formed in 1979 and dominated by the aryregime. TheCPIandGRIevaluatedwithdatataken
FSLN, applied a new model of economic development. from Ref. [19] are displayed in panels (a) and (b) of Fig.
TheleaderofthisadministrationwasDanielJoséOrtega 3, respectively. The results plotted in Fig. 3(b) indicate
Saavedra (Sandinista junta coordinator 1979-85, presi- that the inflation measured in 1989 can be assigned to
dent 1985-90). the beginning of the decrease towards the stable regime.
Economic growth was uneven in the 1980’s [17, 18]. Therefore,we analyzed the hyperinflation using two sets
After the end of the civil war the restructure and re- ofdata,one consideringvaluesofCPI from1969to 1987
build of the economy lead to a positive jump of GDP and the other including the value of 1988also. This sort
of about 5 percent in 1980 and 1981. However, each of fits yielded the parameterslisted in Table I. The solid
year from 1984 to 1990showed a drop in the GDP. Rea- curve in Fig. 3(a) indicates the fit of the shorter series
sons for the contraction included the reluctance of for- to Eq. (2.24). A very steep slope of the CPI may be ob-
eign banks to offer new loans, the diversion of funds to served close to t . If the value of 1988 is included in the
c
fight the new insurrection against the government hold analysis, a piece of information on stabilization is taken
by the Contras. Daniel Ortega began his six-year presi- intoaccount,thenthefittoEq.(2.24)gives,asexpected,
dentialtermonJanuary1985,andestablishedanauster- a largercritical time. This value together with the other
ity program to lower inflation. After the United States parameters and the χ are included in Table I and the fit
Congress turned down continued funding of the Contras is depicted by a dashed curve in Fig. 3(a). For both sets
inApril1985,the Reaganadministrationorderedatotal ofparametersthe GRI is evaluatedwith Eq. (2.21). The
embargo on United States trade with Nicaragua. The results are displayed in Fig. 3(b), where a good match-
United States was formerly Nicaragua’s largest trading ing with measured data may be observed. In addition,
7
1015 8
Nicaragua
Nicaragua
6 (b)
(a)
1010
)
(t) r(t 4
P
105
2 t
c
t
c t
t c
c
100 0
1965 1970 1975 1980 1985 1990 1995 2000 1965 1970 1975 1980 1985 1990 1995 2000
t t
FIG. 3: (a) Squares are yearly CPI in Nicaragua from 1969 to 1997, normalized to P(t = 1969) = 1, presented in a semi-
0
logarithmic plot. (b) Circles are yearly GRI in Nicaragua for the same period as in (a). The solid curve in (a) is the fit of
lnP(t) from 1969 to 1987 with Eq. (2.24), NLF model, while the dashed curve is the fit including the value for 1988. The
solid and dashed curves in (b) stand for r(t) evaluated with Eq. (2.21) of the NLF model, for the shorter and longer series,
respectively. In both drawings, the vertical lines indicate thecorresponding values of t .
c
in both panels of Fig. 3 the vertical lines stand for the data of CPI to Eq. (2.24) in a similar way to that per-
determined values of t . formed by STZ [4]. The obtained parameters and the χ
c
are listed in Table I. A glance at this table indicates a
criticaltimet =2061andasmallexponentofthepower
c
B. A drawback of the NLF model: Israel lawβ =0.069(α≃13). AccordingtoSTZ[4]boththese
values are unrealistic for a developing hyperinflation, in
Let us now focus on the case of Israel. The difficulties addition,they state thatthe resultsare notimprovedby
for determining a reasonable t from data of this coun- reducing the time intervals over which the fits are per-
c
try have a different origin from those found in the cases formed. In addition, they attributed these problems to
of Brazil and Nicaragua. Figure 4(a) shows the yearly the fact that the later prices close to the end of the time
data for the CPI in Israel computed using data taken series start to enter a cross-overto a saturation.
from a Table of the International Monetary Fund (IMF) Furthermore, the values tc = 2061 ± 72 and β =
[13]. The evolution of this CPI may be summarized as 0.069±0.061, which agree with that mentioned by STZ
follows. Deterioration of the internal and external con- [4], were obtained by stopping the minimization proce-
ditions following the energy crisis and the Yom Kippur dure when the variation of χ2 between the i + 1 and
War of 1973 led to an increase in inflation. The labor i iterations was smaller than a standard choice 10−1%.
governmentchoseto accommodateitinthe samewayas However,if one allows to continue the iterations a corre-
Brazil. The single-digit rates of inflation in the 1960’s, lation between these both parameters becomes clear, tc
developed to an annual inflation rate of about 40% in increases while β decreases approaching zero, this hap-
1974-75,about 80% in 1978,and got triple-digit rates of pensinsuchawaythattheproductβ×(tc−t0)converges
about 400% at their peak in the mid-1980’s. Leiderman to a constant yielding a well defined value of the param-
andLiviatan[20]attributedthis responsetothe implicit eter ap given by Eq. (3.5). For instance, in Table I we
preference for short-term considerations of avoiding un- quoted values obtained when the change of χ2 becomes
employment over long-term monetary stability. In 1985 less than 10−3%.
a new strategy was applied that combines drastic cuts Letusnowshowthatbyfollowingtherouteβ →0de-
in government deficit and fixed nominal variables (an- scribed by numerical minimization the NLF expressions
chors), i.e. the exchange rate, wages and bank credit. for GRI and CPI converge to Eqs. (2.16) and (2.17) de-
This approach succeeded in bringing down inflation to a rivedin the MTT’s LF model [10], which correspondsto
moderatelevel(near10%). Inthe90’sinflationtargeting set β =0 in Eq. (2.19). The expressionfor r(t,β →0) is
was adopted and inflation came down to levels recom- obtained starting from Eq. (2.21)
mended by the Organization for Economic Cooperation
1/β
and Development (OECD), i.e., about 2 or 3%. 1
r(t,β →0) = r lim
The hyperinflation since 1969 to 1985 clearly exhibits 0 β→0 1−βa rβ t−t0
a faster than exponential growthas indicated by the up- (cid:20) p 0 ∆t (cid:21)
1/β
ward curvature of the logarithm of CPI as a function 1 (cid:0) (cid:1)
= r lim . (3.9)
of time displayed in Fig. 4(a). In a first step, we fitted 0 β→0(cid:20)1−βap t−∆tt0 (cid:21)
(cid:0) (cid:1)
8
2
STZ*
106 Israel STZ * Israel
(a)
LF ≡ NLF 1.5 (b)
LF ≡ NLF
104 )
P(t) r(t 1
102
0.5
100
0
1970 1975 1980 1985 1990 1970 1975 1980 1985 1990
t t
FIG. 4: (a) Squares are yearly CPI in Israel since 1969 to 1991, normalized to P(t =1969)=1, presented in a semi-logarithmic
0
plot. Thesolid curveindicatesthefitoflnP(t)since1969 to1985 toEqs.(2.17)and(2.24)corresponding totheLFandNLF
models, respectively,whilethedashedcurvestandsforthefitofP(t)toEq.(2.24) asreportedbySTZ*(seetext). (b)Circles
are GRI for the same period as in (a). The solid curve was evaluated with Eqs. (2.16) and (2.21) provided by LF and NLF
models, respectively,while thedashed curvewas computed with Eq. (3.7) of theSTZ* procedure(see text).
3 because ex = ex (see Ref. [21]). Furthermore, imposing
1
the limit β →0 in Eq. (2.23) for CPI one gets
Israel
2 p(t,β →0)=p
0
(t)/P]0 1 ln(r/a)+a(t−t) /∆ t +βli→m0(cid:26)(1r−01−ββ)ap(cid:18)(cid:20)1−βap1r0β t−∆tt0 (cid:21)1−ββ −1(cid:19)(cid:27)
P 0 p p 0 1
n [ 0 =p + r0 lim 1 (cid:0) β(cid:1)−1
ln l 0 ap(cid:26)β→0(cid:20)1−βapr0β t−∆tt0 (cid:21) (cid:27)
r t−t
−1 =p + 0 exp a 0 (cid:0) −1(cid:1) . (3.11)
0 p
LF a ∆t
p (cid:26) (cid:20) (cid:18) (cid:19)(cid:21) (cid:27)
−2 The results obtained in Eqs. (3.10) and (3.11) are equal
1965 1970 1975 1980 1985 1990
to the corresponding formulas of the LF model. There-
fore,wealsofittedtheCPIdatafortheperiod1969-1985
FIG. 5: Squares are yearly CPI data in Israel since 1970 to
directly with LF’s Eq. (2.17). The obtained parameters
1989 taken as lnln[P(t)/P ]. The solid curve is the fit of
0 together with the r.m.s residue χ are included in Table
data since 1969 to 1985 with the complete Eq. (2.17), while
I. The good quality of the fit may be observed in Fig.
thedashedstraight lineistheasymptoticbehavioroftheLF
4(a). Notice the excellent agreement between the values
model, i.e. lnln[P(t)/P ] , given by Eq. (3.12).
0asympt
of r , a , and χ yielded by the LF approach and those
0 p
obtained from the “long” fit with Eq. (2.24) of the NLF
model. Boththesefits areequivalentasindicatedinFig.
Itisnoteworthythatafterthechangeofvariableβ =q−1
4(a). It is also worthwhile to mention that the present
the last expression can be identify with the limit q → 1
value for the parameterutilized in the MTT description,
of the q-exponential function, i.e. ex, used in studies of
q i.e. B = 1+2a = 1.35, is in good agreement with
nonextensive statistical mechanics and economics [21] MTT p
the result 1.4 quoted in Table 1 of Ref. [10]. For the
sake of completeness we plotted in Fig. 4(b) the mea-
r(t,β →0) = r(t,q →1)
sured data of GRI together with the theoretical values
1/(q−1)
1 yielded by Eqs. (2.16) and (2.21) provided by LF and
= r lim
0 q→1(cid:20)1−(q−1)ap t−∆tt0 (cid:21) NLF models, respectively.
TheanalysiswascompletedbyfittingdataofCPIfrom
= r lim e[ap(t−t0)/∆t] (cid:0) (cid:1) 1969 to 1984, i.e. stopping the series before the imposi-
0 q→1(cid:20) q (cid:21) tion of the final stabilization. The results are also in-
t−t0 cluded in Table I, no sizable differences from the fits to
= r exp a , (3.10)
0 p ∆t the larger series were observed.
(cid:20) (cid:18) (cid:19)(cid:21)
9
ThesuccessoftheLF’sdescriptioninthecaseofIsrael izedbyanexponentβ >0,seeEq.(2.18). Inthismodel,
is due to the fact that a double-exponential law is an a critical time t at which the economy would blow up
c
upper bound for data of P(t). This feature is depicted can be determined from a finite time singularity of the
in Fig.5, where measuredvalues oflnlnP(t) areplotted form 1/(t −t)(1−β)/β exhibited by the CPI.
c
togetherwiththefitwiththecompleteLFmodelandthe ItwasfoundthatthehyperinflationoccurredinBrazil
straight line given by the asymptotic expression of this from 1969 to 1994 can be satisfactorily well described
model within the NLF frame if one assumes that, in fact, there
aretwo successiveregimes: one from1969to 1990previ-
P(t) r t−t
lnln =ln 0 +a 0 . (3.12) oustotheCollor’sPlanandtheothersubsequenttothat
p
(cid:20) P0 (cid:21)asympt (cid:18)ap(cid:19) (cid:18) ∆t (cid:19) plan. For the first regime a reasonable tc was obtained.
Thisfeature isinagreementwiththe statementofLucas
One may realize that experimental data of the hyperin-
[14], that parameters can change once policy changes.
flation(i.e. until1985)approachthe asymptoticstraight
Ontheotherhand,theepisodedevelopedinNicaragua
line from bellow.
canbewelldescribedwhenthecorrecteddataareconsid-
Although the LF model provides a good fit, it does
ered. The corrections of the inflation series reported in
not predict any t indicative for a possible crash of the
c the literature are centered around the peak of the data.
economy. In order to estimate a t , the authors of Ref.
c The corrected values yielded a reasonable t within the
c
[4] adopted the same trick as that used in the cases of
frame of the NLF model.
Brazil and Nicaragua, i.e., fitting data of P(t), rather
Finally, by applying the NLF model to the weakerhy-
than values of lnP(t), with the r.h.s. of Eq. (2.24). The
perinflation of Israel no t is got. Moreover, the data
results reported in Table 2 of Ref. [4] are included in the c
are consistent with β → 0 and, in turn, this limit leads
present Table I and the fit is shown in Fig. 4(a). For
to the linear feedback proposed in Ref. [10] which does
the sake of completeness, r(t) was calculated using Eq.
not predict any t . In this case there is neither bifur-
(3.7) corresponding to the STZ* choice and displayed in c
cation of CPI nor correction of data, instead, there is a
Fig. 4(b)). However, this procedure for overhauling the
slowly increasing hyperinflation due to a permanent but
lack of a theoretical tool able to account for any degree
incomplete effort to stop inflation.
of saturation does not preserve the logical structure of
Since it would be of interest to estimate a t within a
the entire model. As emphasized above, P(t) is the ex- c
self-consistent theory even in the case of a weak hyper-
ponential of the integral of r(t) as given by Eq. (2.6).
inflation, we shall propose in a forthcoming work an ex-
In this case r(t) would be given by Eq. (3.7). In turn,
tensionof the NLF model including the effect of a latent
this expression for r(t) should be obtained as a solution
incomplete stabilization. This purpose will be achieved
of a differential equation, e.g. Eq. (2.18), which must
by introducing a new parameter acting on all past r(t).
be formulated in a dynamical description of this kind of
Letusfinishemphasizingthattheselessonsshouldnot
economic system. The latter requirement is not fulfilled
be lost, but instead should be kept in mind to avoid
in the analysis performed by STZ*, hence, it remains as
the repetition of that unpleasing experiences. Moreover,
a simple fit to a selected expression only.
one should always remain the statement of Keynes [22],
namely that: “even the weakest government can enforce
inflation when it can enforce nothing else”.
IV. SUMMARY AND CONCLUSIONS
In the present work we treated regimes of hyperinfla-
Acknowledgments
tion in economy. The episodes occurredin Brazil, Israel,
and Nicaragua were revisited. These new studies indi-
cated that after some management of data outlined in ThisworkwassupportedinpartbytheMinistryofSci-
Sec. IIIA the cases of Brazil and Nicaragua were suc- ence and Technology of Argentina through Grants PIP
cessfully described within the frame of the NLF model 0546/09 from CONICET and PICT 2011/01217 from
availablein the literature [4, 6]. This formalismoutlined ANPCYT, and Grant UBACYT 01/K156 from Univer-
in Sec. IIB is based on a nonlinear feedback character- sity of Buenos Aires.
[1] R.N. Mantegna and E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge
UniversityPress, Cambridge, England, 1999.
[2] S. Moss de Oliveira, P.M.C. de Oliveira, and D. Stauffer, Evolution, Money, War and Computers, Teubner, Stuttgart-
Leipzig, 1999.
[3] D. Sornette, Why Stock Markets Crash (Critical Events in Complex Financial Systems), Princeton University Press,
10
Princeton, 2003.
[4] D. Sornette, H. Takayasu, and W.-X. Zhou, Finite-time singularity signature of hyperinflation, Physica A 325 (2003)
492-506.
[5] M.A. Szybisz and L. Szybisz, Finite-time singularity in the evolution of hyperinflation episodes,
http://arxiv.org/abs/0802.3553.
[6] M.A. Szybisz and L. Szybisz, Finite-time singularities in the dynamics of hyperinflation in an economy, Phys. Rev. B 80
(2009) 0261167/1-11.
[7] L. Szybisz and M.A. Szybisz, People’s Collective Behavior in a Hyperinflation, Advances Applic. Stat. Sciences 2 (2010)
315-331.
[8] L.Szybiszand M.A. Szybisz,Universality of thebehaviorat thefinal stage of hyperinflation episodes in economy, Anales
AFA 26 (2015) 142-147 (in Spanish).
[9] P. Cagan, The monetary dynamics of hyperinflation, in: M. Friedman (Ed.), Studies in the Quantity Theory of Money,
Universityof Chicago Press, Chicago, 1956.
[10] T. Mizuno, M. Takayasu, and H. Takayasu, The mechanism of double-exponential growth in hyperinflation, Physica A
308 (2002) 411-419.
[11] L.C. Bresser Pereira and Y. Nakano, Hyperinflation and stabilization in Brazil: The first Collor Plan, Post-Keynesian
Conference, Knoxville, Tennessee, June 1990; in: P.Davidson and J.Kregel (Eds), Economic Problems of the 1990’s,
Edward Elgar, London,1991 (pp41-68).
[12] M. Mérette, Post-Mortem of a Stabilization Plan: The Collor Plan in Brazil, Journal of Policy Modeling 22 (4) (2000)
417-452.
[13] Table of theInternational Monetary Fund,http://www.imf.org/external/pubs/ft/weo/2002/01/data/index.htm.
[14] R.E.LucasJr,Econometricpolicyevaluation: Acritique,in: K.BrunnerandA.H.Meltzer(Eds),ThePhillipsCurveand
Labor Market vol1 of Carnegie-Rochester Conference Series on Public Policy, North-Holland, Amsterdam, 1976.
[15] P.R.Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw Hill, New York,1969.
[16] W.H.Press, S.A.Teukolsky,W.T. Vetterling, andB.P. Flannery,Numerical Recipesin Fortran 77, Cambridge University
Press, Cambridge, 1996.
[17] J.A.OcampoandL.Taylor,LaHiperinflaciónNicaragüense,en: J.P.Arellano(Ed.),InflaciónRebeldeenAméricaLatina,
CIEPLAN/HACHETTE, Santiago (Chile), 1990 (pp.71-108).
[18] J.A. Ocampo, Collapse and (Incomplete) Stabilization of the Nicaraguan Economy, in: R. Dornbusch and S. Edwards
(Eds),The Macroeconomics of Populism in Latin America, University of Chicago Press, Chicago, 1991 (pp.331-368).
[19] Table of theInternational Monetary Fund,http://www.indexmundi.com/nicaragua/.
[20] L. Leiderman and N. Liviatan, The 1985 stabilization from the perspective of the 1990’s, Israel Economic Review 1 (1)
(2003) 103-131.
[21] C. Tsallis, C. Anteneodo, L. Borland, and R. Osorio, Nonextensive statistical mechanics and economics, Physica A 324
(2003) 89-100.
[22] J.M. Keynes, A Treatise on Money, vols. I-II, Harcourt, Brace and Co., New York, 1930; and The General Theory of
Employment, Interest and Money, Macmillan and Co., London, 1936; both reprinted by D.E. Moggridge (ed.), in: The
Collected Writings of John Maynard Keynes, Macmillan, London, 1973.