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explain the positive relationship between the steady state level of inflation and business cycle inflation volatility observable in the data. In addition, we contribute.

Inflation Level and Inflation Volatility: A Seigniorage Argument* Mikhail Dmitriev1

Erasmus Kersting2

Florida State University

Villanova University

February 2016

Abstract In this paper we contribute to the literature on environments with active fiscal and accommodating monetary policy. We show that this framework is able to explain the positive relationship between the steady state level of inflation and business cycle inflation volatility observable in the data. In addition, we contribute to the theoretical literature by demonstrating that multiple equilibria arise when the elasticity of money demand exceeds the gross real interest rate. Finally, we prove that the monotonic relationship between steady state inflation level and business cycle inflation volatility holds both in the presence of unique as well as multiple equilibria.

Keywords: Inflation; inflation uncertainty; seigniorage; passive monetary policy. JEL Classification Numbers: E31, E41, E52, E62.


We thank Peter Ireland for guidance and suggestions at every stage of the project. All remaining mistakes are our own. 1 Department of Economics. 288 Bellamy Building. 32306 Tallahassee, FL. Email: [email protected] Web: 2 Department of Economics. Bartley Hall, 19085 Villanova, PA. Email: [email protected] Web:

Electronic copy available at:

1 Introduction Empirically there is a consensus that the rate of inflation is correlated with inflation volatility. An incomplete list of references for developed and emerging countries includes Logue and Willett (1976), Grier and Perry (1998), Daal Naka and Sanchez (2005) and Fountas and Karanasos (2007). Price level fluctuations are undesirable for several reasons. As was shown by, for example, Milton Friedman (1976), inflation variability decreases the duration of non-indexed contracts and causes relative price distortions, since indexation is imperfect. To the best of our knowledge, the only formalized explanation of the link between inflation level and volatility has been offered by Ball (1992). In his model, monetary policy dominates fiscal policy, and higher inflation creates uncertainty about the optimal time to disinflate. This causes a positive correlation between the level and volatility of inflation. Our paper generates the result without the need to model an explicit effort to disinflate by the central bank. Instead, we assume that the central bank is not completely independent and seigniorage revenue from printing money is used to balance the government budget.1 This assumption strikes us as plausible given that countries with high rates of inflation are likely to find themselves in that position precisely because monetary policy cannot be conducted independent from fiscal pressures. Our approach is thus one of ’passive’ monetary policy according to the classification by Leeper (1991). While we follow in the tradition of seminal works such as Aiyagari and Gertler (1985) and Sargent and Wallace (1981) in that our model leads to fiscal demands influencing inflation, we differ in two important aspects. On the one hand, we simplify matters by forcing government debt to be zero, which also has the effect of turning the intertemporal budget constraint into a static one. This assumption is isomorphic to the case where a positive nominal debt ceiling has been reached and the monetary authority has to monetize any additional fiscal expenditure shocks. The main benefit of this simplification is the possibility of deriving analytical solutions. On the other hand, we focus on the effect of steady state inflation on inflation volatility, whereas the literature has been mainly concerned with the effect of fiscal shocks on inflation. We contribute to the literature by showing that this environment of passive monetary and active fiscal policy has multiple equilibria when the elasticity of money demand is high. Higher inflation expectations reduce real money balances and force the monetary authority to increase the money supply to generate seigniorage revenue, driving up current inflation. For the equilibrium to be unique, sunspot shocks should increase the expectation of inflation by more than actual inflation, which will make sunspot solutions explosive. But when the elasticity of money demand is high, real money balances are more sensitive to changes in future inflation expectations, and a positive shock to inflation expectations thus leads to a relatively larger decrease in 1

For empirical evidence linking government deficits to increases in seigniorage revenue see, for example, King and Plosser (1985).

2 Electronic copy available at:

real money balances and seigniorage revenue. As a result, current inflation will have to increase to a larger extent. There will exist a cutoff value of elasticity so that inflation expectations will move less than the current level of inflation, leading to non-explosive sunspot solutions and thus to multiple equilibria. This effect is present if the money demand elasticity exceeds the gross real interest rate. It is absent in Leeper (1991) due to his assumption of unit money demand elasticity. We demonstrate that an economy with a higher steady state rate of inflation experiences higher inflation volatility (for a given variance of the exogenous transfers). What is the intuition behind this result? Seigniorage is approximately equal to the product of inflation and real money balances. Real money balances decrease with higher inflation. Therefore, when inflation is high already and the fiscal shock dictates that seigniorage needs to generate an additional one percent of GDP in revenues, inflation has to increase by more, since real money balances

Inflation Change Required to Generate 1 % GDP Revenue

1 % GDP Revenue

Seigniorage in % GDP

1 % GDP Revenue

(the tax base) are small.

Inflation Change Required to Generate 1 % GDP Revenue

Gross Annual Inflation Figure 1: Laffer Curve for Seigniorage and Inflation Consider the Laffer curve for seigniorage and inflation displayed in Figure 1: Gross inflation is displayed on the horizontal axis and seigniorage revenue is depicted on the vertical axis. The Laffer curve flattens out with higher levels of gross inflation. As a direct result, obtaining one additional percent of GDP in seigniorage revenue requires a larger increase in inflation when the latter is already high.


2 The Model The model consists of households that have an endowment of consumption goods and hold money and bonds. They receive transfers from the government that are financed exclusively by seigniorage. The preferences of the representative household are defined over consumption Ct and real money balances

Mt . Pt

Households maximize the expected present discounted utility:


∞ X i=0

 1−b  1−σ Ct+i γ Mt+i β + 1 − σ 1 − b Pt+i i


The intertemporal rate of substitution is governed by σ > 0, and b > 0 represents the inverse of the interest rate elasticity of money demand. The budget constraint of the household in real terms is Ct +

Mt Bt Mt−1 it−1 Bt−1 + = Yt + + + xt Pt Pt Pt Pt


where Mt and Bt are is the household’s holdings of money and private one-period-bonds, respectively. Bonds pay gross interest it , Yt is the household’s endowment of the consumption good and xt is the real value of government transfer. The first order conditions with respect to bonds and money supply yield: Ct−σ  γ

Mt Pt

 = βit Et

 Pt −σ Ct+1 Pt+1





it − 1 . it


Since the government does not produce anything and everybody is endowed with one unit of a representative good, we have the following goods market clearing condition: Yt = Ct = 1.


The government does not issue debt, but instead manipulates the money supply to cover its expenses.2 Asset market clearing thus implies:


Mt − Mt−1 = xt Pt


Bt = 0.


One motivation for this assumption can be the government having reached a nominal debt ceiling.


Finally, the government transfers follow the following stochastic process xt = x + t


Notice that the process is defined for the level - rather than the logarithm - of transfers because we want to consider exogenous shocks to fiscal policy measured in terms of GDP percentage shares. At the steady state equilibrium there is a link between the level of transfers and the rate of inflation. Log deviations from the steady state level of transfers would therefore introduce a connection between the level of inflation and the volatility of shocks, which is important to avoid. We introduce inflation πt =

Pt Pt−1

and real money balances mt =

Mt Pt

to simplify the system

(3), (4) and (6) and obtain:

γm−b t mt −

= 1 − βEt



mt−1 = xt πt

(9) (10)

The system consisting of (8), (10) and (9) fully characterizes the equilibrium.

3 Steady State To compute the steady state we remove the time subscripts and obtain β γm−b = 1 − π m =x m− π

(11) (12)

After substituting m we can express the level of seigniorage through the level of inflation  x=  m=

πγ π−β


πγ π−β


π−1 π

(13) (14)

Even though we defined x and m in terms of inflation here, inflation is not an exogenous variable. Rather, there is a correspondence between all three variables. But for our purposes we will find it convenient to think of high versus low levels of inflation and compute the corresponding levels of real money balances and transfers.


4 Linearization Let us introduce xˆt = xt − x, m ˆ t = log(mt /m). We linearize equations (10) and (9), and modify (8) to obtain the following system: 1 −1 β m xˆt = mm ˆ t − (m ˆ t−1 − π ˆt ) π xˆt = t

−bm ˆ t = Et π ˆt+1 π

(15) (16) (17)

We solve this system consisting of (15), (16) and (17) by using the following method: we iterate equation (16) one period forward and take its expectation in period t, and simplify it using (15) and (17) to obtain: Et m ˆ t+1 = ((1 − b)/π + b/β)m ˆt


The properties of equation (18) are summarized in the following Lemma. Lemma If b ≥ β there is a unique, non-explosive solution given by m ˆ t = 0. If 0 < b < β there is a unique, non-explosive solution given by m ˆ t = 0 for 1 < π ≤ multiple non-explosive solutions for π >

1−b 1−b/β

1−b . 1−b/β

The system has

given by m ˆ t+1 = ((1 − b)/π + b/β)m ˆ t + ut+1 ,

where Et ut+1 = 0. Proof Equation (18) has a unique solution whenever ((1 − b)/π + b/β) ≥ 1. First, let us assume that β ≤ b < 1. Then b/β ≥ 1 and (1 − b/π) ≥ 0. Therefore, ((1 − b)/π + b/β) ≥ 1. Second, let us assume that b ≥ 1. Then ((1 − b)/π + b/β) positively depends on π. It follows that ((1 − b)/π + b/β) ≥ (1 − b) + b/β ≥ 1. Lastly, let us assume b < β. In that case, ((1 − b)/π + b/β) negatively depends on π. Since ((1 − b)/π + b/β) = 1 for π = solutions for π >

1−b , 1−b/β

we have unique solutions for 1 ≤ π ≤

1−b 1−b/β

and multiple

1−b . 1−b/β

In the case of no multiplicity it is straightforward to arrive at π ˆt =

π xˆt , m


whereas the solution with multiplicity is given by π ˆt = b(1 −

π π )m ˆ t−1 − πut + xˆt . β m


The term ut in (20) can be thought of as a sunspot shock related to self-fulfilling expectations. If consumers expect inflation to be high they will choose to hold low real money balances. All


other things equal, lower real money balances require higher levels of inflation for the fiscal authority to raise a given amount of seigniorage revenue. This dynamic gives rise to selffulfilling expectations. The disturbance ut has zero expected value and variance σu2 . In the derivations below we explicitly consider multiplicity cases where σu2 is strictly positive, which adds to inflation volatility. To summarize, inflation reacts positively to the fiscal policy shocks following the seigniorage revenue logic outlined above. Our analytical solutions permit us to obtain expressions for inflation volatility next. σπ,t−1,t =

p Et−1 [(ˆ πt − Et−1 π ˆt )2 ]


For the unique solution we have  σπ,t−1,t = σπ =

π−β πγ

 1b πσ =

π σ m


in case of multiplicity the result is r σu2 +

σπ,t−1,t = π

1 2 σ m2 


At this point we are ready to show that inflation volatility depends positively on the level of inflation regardless of the presence of multiplicity. Theorem σπ depends positively on π. Proof In case of the unique equilibrium, σπ,t−1,t =

π σ. m 

Steady state real balances m depend

negatively on π, so the full expressionq clearly depends positively on π. In case of multiplicity, σπ,t−1,t = π σu2 + m12 σ2 . Steady state real balances m depend negaq tively on π, which in turn means that σu2 + m12 σ2 depends positively on π. As a result, the full expression depends positively on π. Figure 2 shows the variance ratio of inflation and the exogenous fiscal shock as a function of gross steady state inflation. Following our basic intuition outlined in the introduction, a higher steady state level of inflation corresponds to higher inflation volatility. Figure 2a displays the impact of the money demand elasticity: as b decreases inflation volatility grows faster as levels of inflation rise. This result is intuitive, since b corresponds to the inverse elasticity of the demand for real money balances to the nominal interest rate. The nominal interest rate rises and falls with inflation due to the Fisher effect. When the elasticity is high (or b is low), a larger change in inflation is required to generate a given amount of extra seigniorage revenue to cover the fiscal shock, because consumers decrease their real balances holdings to a larger degree. In contrast, with low interest rate elasticity inflation is less volatile because real money balance holdings are less responsive to an increase in the ”inflation tax”.


The three values of b depicted correspond to the three cases discussed in the Lemma above. Ireland (2009) estimates the interest elasticity of money demand and he obtains b = 0.56, which implies the presence of multiplicity in our framework. Figure 2a

Figure 2b


100 b=0.7, σ u =0 b=0.99 b=1.5






60 σ π /σ 

σ π /σ 















1.02 1.04 1.06 1.08 1.1 π

b=0.7, σ u =0 b=0.7, σ u =26σ  b=0.7, σ u =42σ 



1.06 π



Inflation Level and Inflation Volatility for γ = 0.005, β = 0.99 Figure 2b displays the case of b = 0.7 that corresponds to an environment with multiplicity. In this particular example multiple solutions are present for π ≥

1−b 1−b/β


1−0.7 1−0.7/0.99

= 1.024. For

a given steady state level of inflation, an increase in the variance of sunspot shocks σu leads to an increase in inflation volatility. The levels of σu for the dashed and dotted line have been chosen solely for illustrative purposes. They imply an increase of the relative inflation volatility at the intercept (where π = 1.024) by factors 2 and 3, respectively. In addition, for any given σu the positive monotonic relationship between level and volatility of inflation remains intact.

5 Conclusion and Future Research In this paper we show that in an environment where fiscal policy dominates monetary policy a higher inflation level causes higher inflation volatility. At higher rates of inflation consumers hold lower levels of real money balances, which function as tax base for collecting seigniorage. As a result, when the government needs to finance unexpected purchases inflation has to increase more drastically when inflation is already high and real money balances holdings are smaller. We also contribute to the literature by showing that this environment of passive monetary and


active fiscal policy has multiple equilibria when the elasticity of money demand is high. We analytically describe the family of solutions and show that the positive relationship between steady state inflation and inflation volatility holds in unique as well as multiple equilibria.


References [1] Aiyagari, S. R., and Gertler, M. The backing of government bonds and monetarism. Journal of Monetary Economics 16, 1 (1985), 19–44. [2] Ball, L. Why does high inflation raise inflation uncertainty?

Journal of Monetary

Economics 29, 3 (1992), 371–388. [3] Daal, E., Naka, A., and Sanchez, B. Re-examining inflation and inflation uncertainty in developed and emerging countries. Economics Letters 89, 2 (2005), 180–186. [4] Fountas, S., and Karanasos, M. Inflation, output growth, and nominal and real uncertainty: Empirical evidence for the g7. Journal of International Money and Finance 26, 2 (2007), 229–250. [5] Friedman, M. Inflation and Unemployment. Nobel Prize in Economics documents 19761, Nobel Prize Committee, Dec. 1976. [6] Grier, K., and Perry, M. J. On inflation and inflation uncertainty in the G7 countries. Journal of International Money and Finance 17, 4 (1998), 671–689. [7] Ireland, P. N. On the Welfare Cost of Inflation and the Recent Behavior of Money Demand. American Economic Review 99, 3 (June 2009), 1040–52. [8] King, R. G., and Plosser, C. I. Money, deficits, and inflation. In Carnegie-Rochester conference series on public policy (1985), vol. 22, Elsevier, pp. 147–195. [9] Leeper, E. M. Equilibria under ‘active’ and ‘passive’ monetary and fiscal policies. Journal of Monetary Economics 27, 1 (1991), 129–147. [10] Logue, D. E., and Willett, T. D. A note on the relation between the rate and variability of inflation. Economica 43, 17 (1976), 151–58. [11] Sargent, T., and Wallace, N. Some unpleasant monetarist arithmetic. Quarterly Review, Fall (1981).