«D. Sornette a,b,c, H. Takayasu d, W.-X. Zhou a a Institute of Geophysics and Planetary Physics University of California, Los Angeles, CA 90095 b ...»
Finite-Time Singularity Signature of
arXiv:physics/0301007v1 [physics.soc-ph] 6 Jan 2003
D. Sornette a,b,c, H. Takayasu d, W.-X. Zhou a
of Geophysics and Planetary Physics
University of California, Los Angeles, CA 90095
b Department of Earth and Space Sciences
University of California, Los Angeles, CA 90095
de Physique de la Mati`re Condens´e, CNRS UMR 6622 and
Universit´ de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France e d SonyComputer Science Laboratories, 3-14-13 Higashigotanda Shinagawa-ku, Tokyo 141-0022, Japan E-mail addresses: email@example.com (D. Sornette), firstname.lastname@example.org (H. Takayasu), email@example.com (W.-X. Zhou) Abstract We present a novel analysis extending the recent work of Mizuno et al.  on the hyperinﬂations of Germany (1920/1/1-1923/11/1), Hungary (1945/4/30-1946/7/15), Brazil (1969-1994), Israel (1969-1985), Nicaragua (1969-1991), Peru (1969-1990) and Bolivia (1969-1985). On the basis of a generalization of Cagan’s model of inﬂation based on the mechanism of “inﬂationary expectation” or positive feedbacks between realized growth rate and people’s expected growth rate, we ﬁnd that hyperinﬂations can be characterized by a power law singularity culminating at a critical time tc.
Inﬂation is the economic situation in which prices apparently move monotonically upward and the value of money decreases. To classical economics, inﬂation is the undue increase in the supply of credit above the level that is supported by current savings. High inﬂation is always associated with high rates of money supply growth while the relationship is weak for countries with low inﬂation . Thus, ﬁghting high inﬂation requires reducing the growth rate of the money supply.
Inﬂation is one of the few big issues in macroeconomics, together with unemployment, monetary policy, ﬁscal policy, import-export deﬁcits, productivity, government spending and the business cycle, and has been at the forefront of public battles over the past half-century. A good economic policy should strive to achieve a balance between often contradictory requirements: for instance, many economists assume that unemployment tends toward a natural rate below which it cannot go without creating inﬂation. Samuelson and Solow had brought to the U.S. the empirical evidence, ﬁrst compiled by the British economist A.W. Phillips, that there seems to be a tradeoﬀ between inﬂation and unemployment–that is, higher inﬂation meant lower unemployment.
There is thus a long tradition among economists to adopt monetary policy as a way to keep the economy running on high-employment overdrive. Allowing prices to rise seemed the only humane thing to do. Friedman argued however that the unemployment/inﬂation tradeoﬀ was temporary, and he also pointed out that using ﬁscal and monetary policy to avert recessions was a lot harder than it looked. The diﬃculties stem from the fact that policies designed to restrain inﬂation by lowering the level of aggregate demand will tend to depress investment and harm capacity. Improved industrial performance requires a climate conducive to investment and research and development, which in turn depends on, inter alia, high and stable levels of aggregate demand. Business and inﬂation cycles often result from the combination of endogenous interactions (that can lead to incoherence) and of the eﬀects of institutions to contain these tendencies in the economy. The corresponding economic times series can exhibit smooth growth and well-behaved cycles as possible transitory results of the economic processes, but can also allow for intermittent conditions conducive to the emergence of incoherence or turbulence. Institutional factors attempt to act as circuit breakers on the economy. Whenever institutionally determined values dominate endogenously determined values, the path of the economy is broken and an interactive process, which starts with new initial conditions, generates future values. Speciﬁcally, whenever the economy threatens to behave incoherently, these stabilizers, whether built-in or activated by government authority, prevent the economy from continuing on the prior determined path, with the corresponding added complication and possible elements of destabilization. These are important elements in the path
evolution of inﬂation.
In standard economic theory, inﬂation is associated with money supply growth.
At equilibrium, money determines price level and implies equilibrium in markets for other assets. At equilibrium, money demand depends primarily on income and interest rates. But there are several factors keeping money demand unstable, such as ﬁnancial innovations as well expectations. Indeed, one of the major causes of the complexity in stabilizing inﬂation together with other macroeconomic variables is that expectations of producers, consumers and investors may play a key role in the dynamics. Indeed, investment allocations or inﬂation expectations are inﬂuenced by ex-ante values of the risk premia and ex-post returns are rough approximations of these. Thus, “inﬂationary expectation” occurs when people begin to raise prices not because of actual changes in supply or demand or cost or the size of the money supply, but out of fear that some such changes might happen. In the 1990s, when Alan Greenspan, the chairman of the US federal reserve, said that the U.S.
was still suﬀering from the inﬂationary expectations caused by the monetary excess of the 1970s, he was directly addressing the potential for inﬂation caused by “inﬂationary expectations.” When European central banks added liquidity to the gold market in an attempt to prevent an increase in the price of gold from creating concerns about a decrease in the value of the dollar, they were addressing the psychological component of price stability involved in “inﬂationary expectations.” Mathematically, this dynamics translates into sets of coupled nonlinear equations expressing both the competition and delays between expectations and realizations and the presence of positive and negative feedback loops. The complexity of the resulting dynamics stems from the complex nonlinear negative and positive feedback processes intertwining the diﬀerent component of policies.
There are several causes of inﬂation. A prominent origin is wars, which cause the type of inﬂation that results from a rapid expansion of money and credit.
For instance, in World War I, the American people were characteristically unwilling to ﬁnance the total war eﬀort out of increased taxes. This had been true in the Civil War and was also so in World War II and the Vietnam War.
Much of the expenditures in World War I, were ﬁnanced out of the inﬂationary increases in the money supply. If money supply growth and real income are constant, then expected inﬂation rate equals current inﬂation rate (assuming no change in elasticities). This is more or less the standard situation most of the time, as nominal interest rates and inﬂation often move together. In contrast, if people expect an increase in money growth, this then would lead to expect higher inﬂation. And expectation of higher inﬂation raises inﬂation rate even if money growth does not actually increase.
If inﬂation is perfectly anticipated, it entails no cost for creditors and debtors as nominal interest rates incorporate expected inﬂation and nominal wages adjust to oﬀset price increases. But inﬂation devalues the currency and imposes “shoe leather costs”, that are costs of eﬀorts to minimize cash holding (for instance the time and eﬀort in making lots of trips to ATM machines). Prices will be changed more frequently and this imposes “menu costs,” which are the costs of changing prices.
If inﬂation is unanticipated, it induces transfers of wealth from holders of nominal assets to holders of real assets . Suppose for instance that your savings account pays 8% per year and that you expected 4% inﬂation but the realized inﬂation is 7%. You obtain a real interest rate is 1% instead of the 4% that you expected. You are worse oﬀ but the bank is better oﬀ. Unanticipated inﬂation increases risk of gaining or losing wealth and requires more resources for forecasting inﬂation. Unanticipated inﬂation causes confusion about the relative price movements as it could aﬀect some prices sooner than others.
What if the price of oil increases relative to natural gas? Is that a change in relative prices, or a result of inﬂation? If the former holds, consumers should switch from oil to natural gas for heating. If the latter holds, and they switch, then resources are misallocated. More generally, informal accounts of inﬂation’s eﬀects are common, but there are few models which get to grips with the central eﬀects. Partly as a result of this, and partly as a result of many econometric problems, much of the empirical evidence remains unconvincing (see  for an assessment of the various contributions). For all these reasons, a main target of central banks of developed countries in the last decade of the twentieth century has been a low inﬂation .
As we have seen, inﬂation is ﬁrst-of-all an indirect tax leveraged by governments through their (partial) control of the money supply to help them ﬁnance wars or other expenditures. The problem is that inﬂation is not easily controlled due to the dual eﬀect of ﬁnancial innovations and expectations.
Once people start to expect an inﬂation regime, their expectations may lead to strong positive feedbacks that make inﬂation run away. There are several remarkable historical examples of such runaways, called “hyperinﬂation,” such as those that occurred in Germany (1922-1923), Hungary (1945-1946), Latin America in the 1980s and Russia in the recent years. Such hyperinﬂation phases are very costly to society, as there are enormous “shoe-leather” costs, the workers have to be paid more frequently (even daily) and there are rushes to spend the currency before prices rise further. Hyperinﬂation reduce real value of taxes collected, which are often set in nominal terms and by the time they are paid, real value has fallen. Hyperinﬂation leads to large disruptive eﬀects on price and on wage changes and prevents distinguishing relative from aggregate price movements. Wealth allocation becomes very ineﬃcient. Detecting hyperinﬂation in an early stage might contribute to avoid such tragedy.
In a recent work, Mizuno et al.  have analyzed the hyperinﬂations of Ger-
many (1920/1/1-1923/11/1), Hungary (1945/4/30-1946/7/15), Brazil (1969Israel (1969-1985), Nicaragua (1969-1991), Peru (1969-1990) and Bolivia (1969-1985), and showed that the price indices or currency exchange rates of these countries grew super-exponentially according to a double-exponential bt function eb1 e 2 of time (with b1, b2 0). This super-exponential growth was argued to result from a nonlinear positive feedback process in which the past market price growth inﬂuences the people’s expected future price, which itself impacts the ex-post realized market price. This autocatalytic process is fundamentally based on the mechanism of “inﬂationary expectation” alluded to above and is similar to the positive feedbacks occurring during speculative bubbles due to imitative and herd behaviors .
Clearly, a super-exponential growing inﬂation is unsustainable. While providing a useful mathematical description of hyperinﬂation, the double-exponential model of Mizuno et al does not provide a rigorous determination of the end of the hyperinﬂation regime . Here, we re-examine the theory and empirical evidence developed in  and show that the double-exponential law is nothing but a discrete-time approximation of a general power law growth endowed with a ﬁnite-time singularity at some critical time tc. The ﬁnite-time singularity allows us to deﬁne unambiguously the theoretical end of the hyperinﬂation regime as being tc by deﬁnition. This theory provides the ﬁrst practical approach for detecting hyperinﬂation and predicting its future path until its end.
In practice, the end of an hyperinﬂation regime is expected to occur somewhat earlier than at the asymptotic critical time tc, because governments and central banks are forced to do something before the inﬁnity is reached in ﬁnite time.
Such actions are the equivalent of ﬁnite-size and boundary condition eﬀects in physical systems undergoing similar ﬁnite-time singularities. Hyperinﬂation regimes are of special interest as they emphasize in an almost pure way the impact of collective behavior of people interacting through their expectations.
2 From double-exponential growth to ﬁnite-time singularity
In order to establish the correspondence between the double-exponential and the power law formulas, let us ﬁrst summarize the approach followed by Mizuno et al.  who extend Cagan’s theory of inﬂation  in terms of a set of evolution equations linking the market price p(t) with the people’s averaged expectation price p∗ (t). These two prices are thought to evolve due to a positive feedback mechanism: an upward change of market price p(t) in a unit time ∆t induces a rise in the people’s expectation price, and such an anticipation pulls up the market price. Cagan’s assumption that the growth rate of p∗ (t) is proportional to the past realized growth rate of the market price p(t) is expressed by the following equation
whose solution is r(t + ∆t) = r(t − ∆t) and expresses the spontaneous formation of a constant ﬁnite growth rate characterizing a steady state exponential inﬂation.
Cagan’s original model is recovered for the special case b = 1. The system (1,7) or equivalently (5,8) studied by Mizuno et al.  is obtained from a coarse-graining (or Monte-Carlo renormalization group) procedure of a more general system of equations developed by Mizuno et al. .