# Introduction The monthly continuous inflation rate that maxi

Introduction
The monthly (continuous) inflation rate that maximizes the inflation tax revenue varies widely (from 18.3% to 143%), according to semi-elasticity estimates for the German hyperinflation made by several authors (Cagan, 1956; Barro, 1970; Frenkel, 1977; Sargent, 1977; Goodfriend, 1982; Burmeister and Wall, 1987; Christiano, 1987; Casella, 1989; Taylor, 1991; Engsted, 1993; Imrohoroglu, 1993; Michael et al., 1994). Those estimates lead one to conclude that during hyperinflation the German government could have obtained more tax revenue with lower inflation rates. This is an old puzzle of the hyperinflation literature raised by Cagan\'s (1956) seminal paper, namely: that only an economy with an irrational Government could operate on the ‘wrong’ side of the Laffer curve, since the Government could collect more tax with lower inflation rates. The weak hyperinflation hypothesis presented in this paper is consistent with the economy being on the ‘wrong’ side of the Laffer curve, for some time, during hyperinflation, even when agents are rational. Indeed, this “puzzling” outcome is predicted by the weak hyperinflation hypothesis.
In nmda to the traditional theories of hyperinflation, e.g. Sargent and Wallace (1987) and Bruno and Fischer (1990) – which assume a constant fiscal deficit – the driving force to cause hyperinflation in our framework is an increasing fiscal deficit being financed by money (see Barbosa et al., 2006). We understand that this assumption is crucial from an empirical viewpoint since, to our best knowledge, non-constant, increasing fiscal deficits have been ubiquitous in actual hyperinflation episodes. The increasing deficit reaches a point where the intertemporal government budget constraint is not sustainable anymore. This point characterizes the start of the hyperinflation. The process lasts at most a time span before the fiscal policy collapses.
Besides the theoretical framework, this paper departs from other papers in the literature in several ways. First, it follows a different empirical strategy and tests hyperinflation hypotheses estimating the inflation tax revenue curve (inflation tax curve, for short) directly, which can be used to discriminate among different hypotheses. Second, the inflation tax curve functional form used encompasses several specifications as particular cases, making inference more reliable. Thus, this approach allows one to test whether or not the demand for money specification used by Cagan is appropriate. Third, the inflation tax revenue data refer to the 1947–2003 period. This period includes the Brazilian hyperinflation that lasted a very long period, starting in the first half of the 1980s and ended in 1994, with the Real Plan. Therefore, in contrast to other empirical studies, which use very small samples covering only hyperinflation periods, the sample here covers almost half a century, in which both inflation and the inflation tax revenue showed great variability.
The paper is organized as follows: Section 2 presents our hyperinflation model; Section 3 lays out a functional form for the inflation tax curve, which encompasses several specifications and presents graphical evidence on the link between the inflation rate and the inflation tax revenue for Brazil; Section 4 tests the hyperinflation model presented in this paper by estimating the inflation tax curve using cointegration techniques and Section 5 concludes.

Hyperinflation model
This section presents an extension of the rational expectations fiscal crisis model, due to Barbosa et al. (2006). The basic hypothesis of this as well as other hyperinflation models are that money finances the fiscal deficit. Thus, the government flow budget constraint is given by:A dot represents a time derivative. Money creation depends on the price level (P) and the fiscal deficit (f). The real seigniorage is equal to the fiscal deficit. Taking the time derivative of real cash balance (m=M/P) this flow budget constraint can be written as:where τ(m)=πm, the inflation tax revenue, is a function of the real cash balance, the tax base. The inflation rate π is the tax rate. The flow budget constraint states that the change in tax base is equal to the difference between real seigniorage and the inflation tax revenue.