Finite-time singularities in the dynamics of hyperinflation in an economy

Martín A. Szybisz and Leszek Szybisz

Phys. Rev. E 80, 026116 – Published 17 August 2009

Abstract

The dynamics of hyperinflation episodes is studied by applying a theoretical approach based on collective “adaptive inflation expectations” with a positive nonlinear feedback proposed in the literature. In such a description it is assumed that the growth rate of the logarithmic price, $r (t)$ , changes with a velocity obeying a power law which leads to a finite-time singularity at a critical time $t_{c}$ . By revising that model we found that, indeed, there are two types of singular solutions for the logarithmic price, $p (t)$ . One is given by the already reported form $p (t) \approx {(t_{c} - t)}^{- α}$ (with $α > 0$ ) and the other exhibits a logarithmic divergence, $p (t) \approx ln [1 / (t_{c} - t)]$ . The singularity is a signature for an economic crash. In the present work we express $p (t)$ explicitly in terms of the parameters introduced throughout the formulation avoiding the use of any combination of them defined in the original paper. This procedure allows to examine simultaneously the time series of $r (t)$ and $p (t)$ performing a linked error analysis of the determined parameters. For the first time this approach is applied for analyzing the very extreme historical hyperinflations occurred in Greece (1941–1944) and Yugoslavia (1991–1994). The case of Greece is compatible with a logarithmic singularity. The study is completed with an analysis of the hyperinflation spiral currently experienced in Zimbabwe. According to our results, an economic crash in this country is predicted for these days. The robustness of the results to changes of the initial time of the series and the differences with a linear feedback are discussed.

Received 21 January 2009

DOI:https://doi.org/10.1103/PhysRevE.80.026116

Authors & Affiliations

Martín 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, Av. Rivadavia 1917, RA-1033 Buenos Aires, Argentina

^*Corresponding author; szybisz@tandar.cnea.gov.ar

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Vol. 80, Iss. 2 — August 2009

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Figure 1
(a) Yearly GR of the logarithmic price in Peru from 1969 to 1990 (circles). (b) Semilogarithmic plot of the yearly CPI during the same period normalized to $[P (t_{0} = 1969) = 1]$ (squares). The solid curves are four-parameter $GR + CPI$ fits to Eqs. (2.20, 2.24). In both panels the vertical solid line indicates the predicted critical time $t_{c}$ and vertical dashed lines are the error quotes for it, see text for explanations.Reuse & Permissions
Figure 2
(a) Monthly GR of the log-price in Greece from 1941:04 to 1944:10 (circles). (b) Semilogarithmic plot of the monthly CPI in the same period normalized to $P (t_{0} = 1941 : 04) = 1$ (squares). In both panels the open star is the value of 1944:11:10. The horizontal line in (a) is the average growth rate in the period 1941:04–1942:10, while the straight line in (b) is given by Eq. (2.12) in the same period. The solid curves in (a) and (b) are four-parameter $GR + CPI$ fits of data of the period 1943:02–1944:10 to Eqs. (2.20, 2.25) (see text). The vertical solid line is the predicted critical time $t_{c}$ , while the vertical dashed lines indicate its error bars.Reuse & Permissions
Figure 3
(a) Monthly GR of the logarithmic price during the hyperinflation of Yugoslavia from 1990:12 to 1994:01 (circles). (b) Semilogarithmic plot of the monthly CPI normalized to $P (t_{0} = 1990 : 12) = 1$ (squares). The solid curves in (a) and (b) are four-parameter $GR + CPI$ fits of all the data to Eqs. (2.20, 2.24). In both panels, the solid vertical line is the corresponding predicted critical time $t_{c}$ and the dashed vertical lines are its error bars. The dashed and dash-dotted curves are fits of data of the periods 1990:12–1993:12 and 1990:12–1993:11, respectively (see text).Reuse & Permissions
Figure 4
(a) Yearly GR of the log-price in Zimbabwe from 1979:12 to 2007:12 (circles). (b) Semilogarithmic plot of the corresponding CPI normalized to $P (t_{0} = 1979 : 12) = 1$ (squares). The star is the CPI at end of 2008:06. The solid curves in (a) and (b) are four-parameter $GR + CPI$ fits of the data to Eqs. (2.20, 2.24). In both panels, the vertical solid line is the predicted critical time $t_{c}$ and the dashed lines are its error bars.Reuse & Permissions
Figure 5
(a) Monthly GR of the logarithmic price in Zimbabwe from 1998:12 to 2008:07 (circles). (b) Semilogarithmic plot of the corresponding CPI normalized to $P (t_{0} = 1998 : 12) = 1$ (squares). The solid curves in panels (a) and (b) are the four-parameter $GR + CPI$ fits of the data to Eqs. (2.20, 2.24), respectively, while the dashed curves in that panels are three-parameter fits to Eqs. (2.17, 2.19). In both panels, the vertical solid line is the predicted critical time $t_{c}$ and the dashed lines are its error bars.Reuse & Permissions

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