this as an equation for inflation gives us

= -G

Mt) mit)

(9.43)

m(t) mit)

where the second line uses the fact that m{t)gM(t) = G (see [9.42]). Substituting this expression into (9.41) yields

Mt) m(t)

= )s{ln

G m{t) m{t)

-lnm(t)

(9.44)

We can now solve this expression for m(t)/mit); this yields

Mt) mit)

1 - bp

InC - b b

-Inmit)

1 - bp m(t) L

mit) InC -lnm(t)

(9.45)

mit)-G

Our assumption that G is greater than 5* implies that the expression in brackets is negative for all values of m. To see this, note first that the rate of inflation needed to make desired money holdings equal m is the solution to Ce = m; taking logs and rearranging the resulting expression shows

FIGURE 9.9 The dynamics of the real money stock when seignorage needs are unsustainable

"By differentiating (9.45) twice, it is straightforward to show that d inm /(d In ) < 0, and thus that the phase diagram has the shape shown.

Recall that this analysis depends on the assumption that fi < lib. [f this assumption fails, the denominator of (9.45) is negative. The stability and dynamics of the model are peculiar m this case. If G < S*, the high-mflation equilibrium is stable and the low-mflation equilibrium is unstable; ifG > S*,m <0 everj-where, and thus there is explosive deflaUon. And with G in either range, an increase m G leads to a downward jump m inflation (to see this, note that [9.45] implies that the increase leads to an upward jump m th / ; from [9.41], this means that must jump down).

One mterpretation of these results is that it is only because parameter values happen to fall in a particular range that we do not observe such unusual outcomes m practice. A more appealing interpretation, however, is that these results suggest that the model omits important features of actual economies. For example, if there is gradual adjustment of both real money holdmgs and expected inflation, then the stability and dynamics of the model are reasonable regardless of the adjustment speeds. More importantly, Ball (1993) and Cardoso (1991) argue that the assumption that is fixed at omits crucial features of the dynamics of high inflanons (though not necessarily of hyperinflations). Ball and Cardoso develop models that combine seignorage-driven monetary policy with the standard Keynesian assumption that aggregate demand pohcies can reduce inflation only by temporarily depressmg real output. They show that with this assumption, only the low-mflation steady state is stable. They then use their models to analyze a variety of aspects of high-inflation economies.

that this inflation rate is (InC ~lnm)/b. Next, that if real money holdings are steady, seignorage is ; thus the sustainable level of seignorage associated with real money balances of m is [(In -lnm)/b]m. Finally, recall that S * is dehned as the maximum sustainable level of seignorage. Thus the assumption that S* is less than G implies that [(In -Inm)/b]m is less than G for all values of m. But this means that the expression in brackets in (9.45) is negative.

Thus, since bp is less than 1, the right-hand side of (9.45) is everywhere negative: regardless of where it starts, the real money stock continually falls. The associated phase diagram is shown in Figure 9.9. With the real money stock continually falling, money growth must be continually rising for the government to obtain the seignorage it needs (see [9.42]). In short, the government can obtain seignorage greater than S *, but only at the cost of explosive inflation.

This analysis can also be used to understand the dynamics of the real money stock and inflation under gradual adjustment of money holdings when G is less than S*. Consider the situation depicted in Figure 9.8. Sustainable seignorage, ttzti *, equals G if inflation is either gi or gz; it is greater than G if inflation is between gi and gz] and it is less than G otherwise. The resulting dynamics of the real money stock implied by (9.45) for this case are shown in Figure 9.10. The steady state with the higher real money stock (and thus with the lower inflation rate) is stable, and the steady state with the lower money stock is imstable.

This analysis of the relation between seignorage and inflation explains many of the main characteristics of high inflations and hyperinflations. Most basically, the analysis explains the puzzling fact that inflation often reaches

FIGURE 9.10 The dynamics of the real money stock when seignorage needs are sustainable

extremely high levels. The analysis also explains why inflation can reach some level-empirically, in the triple-digit range-without becoming explosive, but that beyond this level it degenerates into hyperinflation. In addition, the model explains the central role of fiscal problems in causing high inflations and hyperinflations, and of fiscal reforms in ending them (Sargent, 1982; Dornbusch and Fischer, 1986).

Finally, the central role of seignorage in hyperinflations explains how the hyperinflations can end before money growth stabilizes. As described in Section 9.2, the increased demand for real money balances after hyperinflations end is satisfied by continued rapid growth of the nominal money stock rather than by declines in the price level. But this leaves the question of why the pubhc expects low inflation when there is stiU rapid money growth. The answer is that the hyperinflations end when fiscal and monetary reforms eliminate either the deficit or the governments ability to use seignorage to finance it, or both. At the end of the German hyperinflation of 1922-23, for example, Germanys World War 1 reparations were reduced, and the existing central bank was replaced by a new institution with much greater independence. Because of reforms like these, the public knows that the burst of money growth is only temporary (Sargent, 1982).

To incorporate the effects of the knowledge that the money growth is temporary Into our formal analysis, we would have to let the change in real money holdings at a given time depend not just on current holdings and current inflation, but on current holdings and the entire expected path of inflation. See n. 36.