Discussion Papers In Economics
And Business

Seigniorage Revenue or Consumer Revenue?
Theoretical and Empirical Evidences
Tatsuyoshi Miyakoshi
Discussion Paper 08-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSeigniorage Revenue or Consumer Revenue?
Theoretical and Empirical Evidences
Tatsuyoshi Miyakoshi
Discussion Paper 08-11

March 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSeigniorage Revenue or Consumer Revenue?
Theoretical and Empirical Evidences
By
Tatsuyoshi Miyakoshi*
Osaka University
March 2008

Abstract
The purpose of this paper is to propose a seigniorage model including the contributionsof Bailey (1956) and Marty (1976), using a different framework to Mankiw (1987), totest whether their results are supported, and use a numerical example to estimate theseigniorage model. The government decides the money growth rate to maximize thesocial welfare function for each of seigniorage revenue aversion, loving and neutrality.
The numerical example using Japanese data shows that the social welfare function
supports seigniorage revenue aversion, supporting the results of Bailey and Marty, andthat the degree of seigniorage revenue aversion is stronger in the 2000s than in the
1990s.

Keywords: Bailey and Marty; Social welfare function; Mankiw model; NumericalexampleJEL Classification Number: E40; E50; C40; C50* Earlier versions of this paper were presented at the 2007 Japan Society of Household
Economics (Okinawa, Japan) and the 2007 Western Economics Association Conference(Seattle, USA). I am very grateful to Osamu Kamoike, Yoshinori Kon, Victor Li, andYoshiro Tsutsui and participants at the conferences for their helpful comments. The
research of the author was supported by Grant-in-Aid 16530204 from the Ministry of
Education, Culture, Sport, Science and Technology of Japan.
Correspondence: Tatsuyoshi Miyakoshi, Prof. of Monetary Economics,
Osaka School of International Public Policy, Osaka University , 1-31, Machikaneyamamachi, Toyonaka, Osaka, 560-0043, Japan.
tel:+81-6-6850-5638; fax:+81-6-6850-5656, E-mail: miyakoshi@osipp.osaka-u.ac.jp1

1.

Introduction
Money growth theory has two strands. First, the optimal money growth rate aims tocontrol inflation, GDP, the exchange rate and so on. Second, the optimal growth rate
aims to maximize seigniorage revenue. We consider the second strand in this paper. To
date, the so-called seigniorage model in which the government decides the money
growth rate to maximize seigniorage revenue was developed by Bailey (1956) andextended by Blanchard and Fischer (1989, p. 188–194). Honohan (1996), Loviscek(1996), Turner and Benavides (2001) and Tekin-Koru and Özmen (2003) have applied
this model to empirical studies. However, prior to Blanchard and Fischer (1989), Bailey(1956) and Marty (1976) proposed a seigniorage model where the government
maximizes the seigniorage revenue together with consumer revenue. This is because
seigniorage revenue reduces consumer revenue and induces a welfare loss. Then, basedon the Bailey and Marty models, Mankiw (1987) proposed a seigniorage model where
the government chooses the rates of tax and money growth to minimize the social costof raising revenue from tax and seigniorage. Amano (1998) and Ho (2003) provided
support for Mankiw’s model.
The purpose of this paper is to propose a seigniorage model including thecontributions of Bailey and Marty, using a different framework from Mankiw (1987), totest whether their results are supported, and use a numerical example to estimate theseigniorage model. The government decides the money growth rate to maximize thesocial welfare function involving seigniorage revenue aversion, loving and neutrality.
The numerical example using Japanese data supports the existence of a social welfare
function consistent with seigniorage revenue aversion, supporting the results of Bailey
and Marty, and shows that the degree of seigniorage revenue aversion is stronger in the
2000s than in the 1990s. This paper is organized as follows. In Section 2, we propose aseigniorage model. In Section 3, we propose a statistical methodology of estimating andtesting this model. In Section 4, we present a numerical example to evaluate whether thesocial welfare function supports seigniorage revenue aversion (Bailey and Marty’s
result) and the magnitude of the degree of aversion. Section 5 concludes the paper.

2.

The Model

2.1.

Market equilibrium
We briefly present the seigniorage model of Blanchard and Fischer (1989, p.
188–194), which is composed of two equations dealing with the relationship between
money and price, and transform their model to a discrete type from a continuous one forthe sake of empirical analysis:
mt ≡Mt= A exp(− aRt ) y , A > 0, a > 0, Rt ≥ 0 ,Pt

π te+1 − π te = b(π t − π te ), b > 0, ab ≠ 1 ,

(1)

(2)

2

where Rt = r + π te+1 ( r is the real interest rate and πt+1e ≡ (Pet+1 – Pt)/Pt is the expected
inflation rate formed at period t) is the nominal interest rate, y is real output, Mt is themoney supply, Pt is the price level and A, a, b are positive constant terms. Furthermore,πt ≡ (Pt – Pt–1)/Pt–1 is the actual inflation rate observed at period t. The real variables
r , y are assumed to be constant over all periods. Equation (2) shows the expected
inflation πt+1e (price Pet+1) formed at period t, given the actual inflation πt at period t andexpected inflation πet at period t–1. Equation (1) shows that the actual inflation πt (pricePt) clears the money market. The government controls only the constant money growthrate θ over time, θ ≡ ( M t − M t −1 ) / M t −1 .1
How is the inflation rate πt decided by the growth rate θ ? Take logarithms of(1) and rearrange, considering period t and t–1 in equation (1) to obtain:

θ − π t = − a (π te+1 − π te ) .

(3)

Moreover, we insert equation (2) into (3):θ − π t = − ab(π t − π te ) .

(4)Thus, we can get the actual inflation rate πt (i.e., Pt) at period t, given the expected
inflation rate πte at period t–1 and the money growth θ (i.e., Mt). However, given πt–1
and πt–1e, the expectation πte is formed by (2). What are the dynamics of the expectation
πte? Furthermore, what are the dynamics of the actual inflation rate πt? The detailed
derivation and the stability condition are given in Appendix 1. Here, we focus on thesteady state. The steady state of πte = πt+1e lead to πte = πt because of (2), which leads to
πt = θ because of (4) and then πte = θ. The following is true at this steady state: πt+1e = θ
and then Rt = r + π te+1 = r + θ .

2.2. On seigniorage revenue and consumer revenue
The seigniorage at t, i.e., the revenue raised from the creation of money, isdefined as:
St ≡M t +1 − M t M t +1 − M t M t=•= θmt .
Pt
Mt
Pt

(5)

The seigniorage revenue S at the steady state (where θ = πte = πt, and money demand (1)equals supply) is, considering (1):S = θ Ay exp(− a(r + θ )) ≡ g (θ ) ,(6)dS / dθ = Ay (1 − aθ ) exp(−a(θ + r )) .
1

The validity of these assumptions was explained by the model of Sidrauski (1967) and further
developed by Blanchard and Fischer (1989, p. 188–198).

3

Here, the subscript t denoting the period is deleted as we consider the steady state
hereafter. The seigniorage revenue S is illustrated as the shaded part in Figure 1. In theseigniorage model of Blanchard and Fischer (1989), the maximized seigniorage revenue
S* of (6) at the steady state is obtained at:θ* = 1/a.: S* = S(θ*).

(7)

We assume that a consumer decides money demand m to maximize theconsumer profit V, given R:
V ≡ U (m) − mR ,(8)

where U (m) is a consumer’s utility measured in money m, the so-called consumer
revenue, and mR is the cost of interest payments, the so-called consumer cost. The
consumer’s maximum profit V is obtained at md where U′(md) = R. Its maximum valueis:md∫ U ′( x)dx − m R .
d

(9)

0

On the other hand, the bank profit is assumed to be the interest payment Rmd. Then, theprivate sector profit is the total profit of the consumer and the bank, which is always
equal to the consumer revenue as long as the consumer behaves so that:

∫

md

0

md

U′(x)dx − md R + md R = ∫ U′(x)dx .

(10)

0

Using the necessary condition for the maximum, U′(md) = R, we can rewrite (10) as
md

md

0

0

∫ U′(x)dx = ∫ R(x)dx . Moreover, the consumer revenue U(m ) is expressed asdU(m), considering that md equals the money supply mS, which is exogenously controlled
by the government, through the adjustment of R because of the money market
equilibrium condition (1): md = mS = m. Thus, given the exogenously determined money
supply (mS = m) controlled by the government, the maximized private sector profit(which equals consumer revenue) is as follows:m ⎡ ln( Ay )1m(ln( Ay) +1− ln(m))⎤− ln(x)⎥dx =U (m) ≡ ∫ ⎢
0
a
a
⎣ a
⎦
Ay(a(θ + r ) +1)exp(−a(θ + r ))=≡ f (θ ), ,adU / dθ = Ay(−a(θ + r ))exp(−a(θ + r ))where R =

(11)

1
1
ln( Ay ) − ln(m) is resolved in terms of R after taking the logarithm of (1).
aa4Next, we consider how the government maximizes the private sector profit by adjusting
the exogenously given money supply m. By using (11), the maximization of total
private profit is obtained at:θ ∗∗ = – r , that is, R∗∗ = θ ∗∗ + r = 0 : U ∗∗ = U (θ ∗∗ ) .

(12)[INSERT Figure 1]

2.3.

Government behavior
The government chooses the money growth rate θ to maximize the following
social welfare function W, which is expressed in terms of the exogenously given money
supply m (θ ) controlled by the government:

Max W = (S + S )αU1−α :θS = g(θ ), U = f (θ ), S + S > 0,U > 0, 0 ≤ α ≤1, S > 0

,

(13)

where the social welfare function consists of seigniorage S and consumer revenues(private sector profit) U, considering the viewpoint of Bailey (1956) and Marty (1976).
The seigniorage revenue S may be negative if the growth rate θ is negative, as is
obvious in (6). The minimum of S is − S at θ = −r , i.e., R = θ + r = 0 . Then,
S = (−1)(− r ) Ay exp(− a(− r + r )) = rAy . This social welfare function W has thefollowing features, as shown in Figure 2. In particular, when the function displays
seigniorage revenue loving, that is α = 1, it is flat against the U axis and social welfare
increases only through an increase in seigniorage revenue. However, when the function
displays seigniorage revenue neutrality, that is α = 0, it is flat against the S axis andsocial welfare increases only through increases in consumer revenue. Finally, when thefunction displays seigniorage revenue aversion, that is 0 < α < 1, it is convex against theorigin. The decrease in α produces an increase in seigniorage revenue aversion.
[INSERT Figure 2]

We rewrite the maximizing problem (13) as follows, by deleting the money
growth rate θ:

Max W = (S + S )α U1−α subject to S = g( f −1(U)) .
S, U

(14)

S = g(f–1(U)) shows a locus of realizable values of S and U for any θ. We choose S andU through θ. However, in (14), we seem to be choosing S and U directly. The shape of
its locus is as follows. The maximized S is obtained at U = U(θ*) where θ* = 1/a as5

seen in (7), while the maximized U is obtained at S = S (θ ∗∗ ) = S where θ ** = −r asseen in (12), as shown in Figure 3. When θ increases towards θ* = 1/a beyond θ** = –r , U decreases from U ∗∗ = U (θ ∗∗ ) because of (12) and S increases towards S(θ*)
because of (7). When θ increases beyond θ* = 1/a, U decreases because of (12) and S
decreases from S(θ*) because of (7), as shown in Figure 3. Then, U is not a one-to-one
correspondence in S, but U in the region of [θ**, θ*] is a one-to-one correspondence in
S. Thus, S = g(f–1(U)) can be shown in Figure 3.
The derivatives of S = g(f–1(U)) in terms of U are, by using (6) and (11):d S dS1
Ay (1 − aθ ) exp(−a (θ + r ))1 − aθ,==•=Ay (−a (θ + r )) exp(−a (θ + r )) − a (θ + r )dU dθ dUdθd U dU1− a (θ + r )
,
=
•
=
dS
dθ dS
1 − aθ
dθ

(15)

(16)

where R = r + θ > 0 and then θ > −r : the derivative is undefined for θ = − r . What is
the shape of S = g(f–1(U)) in terms of U? Is it strictly concave?

d(dS) / dU =
dU

dS
)
2
1
dU 1 = a r + a<02
2d θ dU a (θ + r ) Ay (− a (θ + r )) exp(−a (θ + r ))
dθ

d(

(17)

Then, S = g(f–1(U)) is strictly concave in U.
Among realizable couples of (S, U) on S = g(f–1(U)) for a given θ, we mustchoose the particular couple of (S, U) to maximize the social welfare function,
W = ( S + S )α U 1−α . When this function displays seigniorage revenue loving (α = 1), themaximized social welfare W is uniquely obtained at point A (S(θ*), U(θ*)) because of
the strict concavity of S = g(f–1(U)) and the convexity of social welfare function W ,where θ* = 1/a, as shown in Figure 3. However, when the function displays seigniorage
revenue neutrality (α = 0), the maximized social welfare W is obtained at point B(S( θ ** ), U( θ ** )) uniquely, where θ ** = −r : i.e., R = θ ** + r = 0 . When the function
displays seigniorage revenue aversion (0 < α < 1), the maximized social welfare W is
obtained uniquely at some point C (S( θ *** ), U( θ *** )) between A and B.

Proposition 1: The optimal θ *** is expressed as follows when 0 < α < 1.

a 2θ *** (θ *** + r )α =1 + ar(18)

6

The proof is given in Appendix 2. Equation (18) can be interpreted as a solution for the
optimal money growth θ *** , given the other parameters. Alternatively, the unknown
preference parameter α is expressed by the observed (optimal) money growth θ,
observed real interest rate r and the estimated money demand parameter a. We havealready shown that the decrease in α (which means that the indifference curve of asocial welfare function in Figure 2 approaches the vertical line) is an increase in
seigniorage revenue aversion.
The actual seigniorage revenue and consumer revenue data, dependent on themoney growth rate θ, are observed in Figure 4. We investigate whether these data areobserved around point A, point B or point C. That is, whether the government has asocial welfare function W displaying seigniorage revenue loving, neutrality or aversion,or, how is the seigniorage model?
[INSERT Figure 3 and Figure 4]

3.

Statistical Methodology
We show a testing and estimation method to evaluate whether the social welfare
function displays seigniorage revenue aversion, as suggested by Bailey and Marty, andestimate the degree of aversion. To do this, we proceed as follows. First, we estimate
the money demand function (1). By using the estimated parameter a, we calculateθ* = 1/a, seigniorage revenue and consumer revenue, i.e., point A (S(θ*), U(θ*)).
However, we calculate point B (S(θ**), U(θ**)) where θ ** = −r , by using
R = θ ** + r = 0 . Second, we test whether the observed data have originated from point
A, B or C. However, because the observed data consists of the growth rate of money θand nominal interest rate R, we test whether the observed money growth rate θ was
obtained from θ* = 1/a and whether the observed nominal interest rate R was obtained
from R = 0. If the data originated from C, this supports the results of Bailey and Marty
that the government considers both revenues. Moreover, by estimating the preference
parameter α in (18) based on the observed data, we measure and compare the degree ofaversion on seigniorage revenue in two different periods.
We estimate the parameter a for money demand function (1) in the logarithm
form:

ln(mt ) = C − aRt , C ≡ ln(Ay) .

(19)

This money demand function is a function at the steady state for not only real, but alsomonetary variables. We then consider the function to be a cointegration term in a twodimensional VAR model with a nominal interest rate R and a real money supply m. We
have to estimate a in (19) as a cointegrating term in the following equation (21). The
following statistical methodology is used. Let Xt = ( ln(mt ), Rt )' and assume that thisvector is generated from a vector autoregression VAR(k) model with a constant term ψand Gaussian errors εt:7Xt =Π1Xt−1 +Π1Xt−2 +...+Πk Xt−k +ψ +εt .

(20)

We write the model in error correction form as:∆Xt = ΠXt −1 +Γ1∆Xt−1 + ... +Γk−1∆Xt−k+1 +ψ + εt :Π ≡Ι −Π1 −Π2 −.... −Πk

.

(21)

The parameters (Γ1,…, Γk–1) define the short-run adjustment to changes in the process,whereas Π = αβ′ defines the short-run adjustment (α) and the long-run relations(β).Johansen and Juselius(1990,1992) shows that if Xt ~ I(1), Π (p × p matrix) has the
reduced rank of r (< p) and can be represented as Π = αβ′. The parameterization inΠ = αβ′ facilitates the investigation of the r linearly independent stationary relations
between the levels of the variables, and the p–r linearly independent nonstationary
relations. Thus, the representation of Π = αβ′ implies that the process ∆Xt is stationary,Xt–1 is nonstationary, but also that β′ Xt–1 is stationary. Thus, we can interpret therelation β′ Xt–1 as the stationary relations among nonstationary variables, i.e., ascointegrating relations. Johansen and Juselius (1990, 1992) developed the likelihood
procedure for estimating the parameters, and testing the order of cointegration rank andthe various hypotheses on the restrictions of parameters.

4.

A Numerical Example

4.1. Data
The monthly data for this study are taken from International Monetary Fund’sInternational Financial Statistics CD-ROM July 2006, and covers the period 1990:1–
2005:12. The prices P are line 63 (consumer price, 2000 = 100) or line 64 (wholesale
price, 2000 = 100), and the nominal interest rate R is line 60P (lending rate %). An
exception is the money supply, which is high-powered money compiled from the Bankof Japan (monetary base, seasonally adjusted, 100 million yen, average outstanding):
HmoneyC (money/consumer price) and HmoneyW (money/wholesale price) are in
logarithm form. The choice of variables followed previous papers including Phylaktis
and Taylor (1993), Loviscek (1996) and Turner and Benavides (2001). Figure 5 shows
real output y, line 66..czf (industrial production seasonally adjusted 2000 = 100), whichshows that output seems to be fixed or stationary as assumed in Section 2. Finally, we
also use deposit money (line 59MBFZF (M2+CDs, seasonally adjusted, 100 millionyen)): DmoneyC (money/consumer price) and DmoneyW (money/wholesale price) arein logarithm form. All data are shown in Figure 5.
We now discuss the difference between the two kinds of money supply, deposit
money and high-powered money. The former is related to consumer revenue U, thelatter to seigniorage revenue S. When the money multiplier between deposit money andhigh-powered money is constant, the growth rate of both measures is equal to each other.
Then, (14) can be theoretically expressed by the same growth rate θ. However, as
shown in Figure 5, the growth rate of both measures is actually different. Moreover, the8money demand function of consumers is for deposit money. Therefore, we use deposit
money and high-powered money as the money supply, as done by previous studies
including Phylaktis and Taylor (1993), Honohan (1996), Loviscek (1996), Turner andBenavides (2001) and Tekin-Koru and Özmen (2003).
[INSERT Figure 5]

4.2. Estimation for the money demand function
We have to estimate the money demand function in (19) as a cointegrating termin (21). From the estimated parameter a in all pairs of cointegration relations, we seek
an optimal money growth θ* = 1/a. The first step is to implement the augmented
Dickey–Fuller (ADF) unit root test (using an autoregression (AR)) for R and the money
supply, and the Johansen–Juselius cointegration test (using the error correction model(ECM) of (21)). A lag length must be chosen for the AR and the ECM following
Phillips (1987) and Gonzalo (1994).2 For the AR, the optimal lag lengths together withthe t-statistics of the ADF tests are reported in Table 1 and all are I(1).3 For the ECM of(21) in Table 2, we found that the two-order to six-order lags satisfied their criteria. The
resulting cointegration rank between R and money is one at the 1% significance level,
according to the λ-max test and the trace test in Table 3. We conclude that all pairs oftime series variables have one cointegrating vector. We have estimated a money
demand function in the cointegrating term, which is shown in Table 4. The optimalmoney growth rate θ* to maximize the seigniorage revenue S is 1/a = 3.08% for
HmoneyW, 2.26% for HmoneyC, 11.09% for DmoneyW and 9.17% for DmoneyC.
Thus, we have estimated two types of money demand function and optimal money
growth, based on consumer and wholesale prices. The difference between deposit andhigh-powered money causes the difference between optimal money growth forseigniorage revenue. The computations have been performed with the computer
package CATS in RATS by Hansen and Juselius (1995).
[Insert Table 1, Table 2, Table 3 and Table 4]2

Our procedure for choosing the optimal lag length was to test between a two- and an 18-order lag
for the AR and between a two- and a six-order lag for the ECM, by using the minimum value of
Schwarz’s Bayesian Information Criterion (SBIC). The residuals from the chosen AR or ECM werethen checked for whiteness. Phillips (1987) and Gonzalo (1994) suggest the robustness of theJohansen–Juselius procedure to heterogeneity and nonnormality. Here, whiteness is checked only by
Ljung–Box (LB) Q tests for absence of correlation for all 18 lags (for AR) and six lags (ECM) at the
5% significance level. If the residuals in any equation proved to be nonwhite, we sequentially chosea higher lag structure until they were whitened.
3
For some data based on visual inspection, we implement Perron’s (1989) unit root test with a breakat 1995:7 for the lending rate R, and at 2003:12 for high-powered money m. The former is I(0), but
the latter is I(1). Not all variables are necessarily I(0). Then, we assume all data are nonstationary
even with a break.

94.3. Testing whether the observed data have occurred around point A, B or C
If the observed money growth rate θ, which is assumed to follow the normal
distribution, occurs around the optimal rates θ* = 1/a, the observed data have occurredat point A and then the social welfare function displays seigniorage revenue loving. We
test this hypothesis as follows. If the null hypothesis is H0: θ = θ* (in Table 5), the test
statistics follows the t-distribution: t = (θ A − θ *) / σˆ ~ t (n − 1) , where n is the number ofobservations in the sample, θ A is the sample mean and σˆ is the standard deviation of
the sample. As shown in Figure 5, the seven high-powered money growth rate data
overshoot 4% in magnitude. We omit these abnormal data prior to testing. The null
hypothesis is rejected for each money demand function at the 5 or 10% significancelevel. When we estimate the money demand function by using HmoneyC, θ*=2.26%
and we have to test this null hypothesis by using the average of the nominal highpowered money growth rate θ A =0.46, as shown in Table 5.
Similar logic is applied to whether the observed data occurs at point B. The null
hypothesis is H0: R = 0 (in Table 6). This hypothesis is rejected at the 10% significancelevel. As a result, the social welfare function displays neither seigniorage revenue
loving by suggesting point A nor seigniorage revenue neutrality by point B. That is, thesocial welfare function displays seigniorage revenue aversion by point C. This resultshows that the Japanese government has a social welfare function W with a view ofseigniorage revenue and consumer revenue that supports Mankiw (1987), where thegovernment chooses the rates of tax and money growth to minimize the social cost ofraising revenue from tax and seigniorage.
[Insert Table 5 and Table 6]

4.4. Degree of seigniorage revenue aversion
We find that the social welfare function displays seigniorage revenue aversion,
supporting Bailey and Marty. The next logical step is to investigate the degree to which
the money growth rate policy with seigniorage revenue aversion changed during theperiod 1990–2005 using the preference parameter α in the social welfare function.
Considering (18), the parameter α is computed by using the average level of theobserved (optimal) money growth θ, the average level of observed real interest rate r ,and the estimated money demand parameter a. Here, the real interest rate r can becomputed as r = R A − θ A , where RA is the average of the observed nominal interest rate,
θA is the average of the observed (optimal) money growth rate and the preference
parameter α is given in Table 4. Thus, as shown in Table 7, by using HmoneyW, thepreference parameter α = 0.083 in the 1990s (1990:1–1999:12) decreases to 0.076 in the
2000s (2000:1–2005:12) by 0.007: that is, a 10% decrease. By using the other money
measures, we get the similar decrease in the preference parameter α. This shows that the10money growth rate policy in the 2000s has stronger seigniorage revenue aversion
compared with the 1990s, as shown in Figure 6.
[Insert Table 7 and Figure 6]

5.

Concluding Remarks
The purpose of this paper was to develop a seigniorage model that includes thecontributions of Bailey and Marty, using a different framework to Mankiw (1987), totest whether their result is supported, and provide a numerical example to estimate theseigniorage model. The government decides the money growth rate to maximize thesocial welfare functions that display seigniorage revenue aversion, loving and neutrality.
The numerical example using Japanese data showed the existence of seigniorage
revenue aversion, supporting Bailey and Marty, and showed that the degree ofseigniorage revenue aversion was stronger in the 2000s than the 1990s. However, thispaper has only provided a theoretical model and an empirical application of the model.
Further empirical studies are needed.

11Appendix 1
Derivation of difference equation and stability condition: Inserting (4) into (2), we
obtain the difference equation:

π te+1 − (1 + µ )π te = − µθ

; µ≡

b
.
ab − 1(A1)

Because 1 + µ ≠ 1 because of (1) and (2), the solution is:
1 − (1 + µ )t
π = c(1 + µ ) +(− µθ ) .
1 − (1 + µ )
e
t

(A2)

t

When |1 + µ | < 1, that is, –2 < µ < 0, the solution is stable. First, we analyze theinterval of the parameters of a and b for µ < 0. Then, ab < 1. Second, for –2 < µ, we
find (2 – b)/2b > a. We require a close interval of both:
1 1
− >a.
b 2

(A3)

This is the stability condition. However, we do not check empirically whether thiscondition holds for observed data. Our model is a difference equation and the stability
condition is different from Blanchard and Fischer (1989, p. 188–198): ab < 1. The
steady state is lim π te = θ . Furthermore, the dynamics of πt are, considering (4):
t →0

πt =

ab e
θ
.
πt −
ab − 1ab − 1(A4)

Because the dynamics of πt do not include t except for πet, the stability condition is thesame as (A3). In addition, the steady state of πt is obviously θ, considering (A4).
■

Appendix 2
Proof of Proposition 1: Taking the logarithm of W in (13) and considering (6) and(11):

ln W = α ln( S + S ) + β ln U
⎛ Ay ( x + 1) exp(− x) ⎞= α ln (θ Ay exp(− x) + rAy ) + (1 − α ) ln ⎜
⎟
a
⎝
⎠
x
= Ay + α ln[( − r ) exp(− x) + r ] + (1 − α )(− ln a ) + (1 − α )(ln( x + 1) − x),axwhere x ≡ a(θ + r ) , θ = ( − r ) and S = rAy
a

12

(A5)

We maximize lnW in terms of x (i.e., maximize W in terms of θ). We take derivatives interms of x:
1 − x + arexp(− x)()
d ln W1a=α+ (1 − α )(− 1) = 0 ,
x
dx
x
+
1
( − r ) exp(− x) + r
a.
d 2 ln W
<
0
dx 2

(A6)

(A7)

Then, when W displays seigniorage revenue loving (α = 1), the optimal money growthrate θ is, considering (A2), 1 − x + ar = 0 , that is, θ = 1/a. Also, when W displays
1
seigniorage revenue neutrality (α = 0),= 1, i.e.,θ = − r . When 0 < α < 1, what is
x +1
the optimal money growth rate θ? In (A6) , we assume exp(− x) ≈ 1 . Because nominalinterest rate R(= θ + r ) and a are empirically zero point something shown in Table 4
and then x= a(θ + r ) is very small. Therefore, (A6) can be rewritten as:
1 − x + ar()
d ln W1a=α+ (1 − α )(− 1) = 0
x
dx
x
1
+
.
( −r)+ra2x − xar α − α (1 + ar ) = 0(A8)

We can solve for x from (4), considering x > 0:1
x = [ar ± a 2 r 2 + 4α (1 + ar ) ]
.
2

(A9)

Due to x = a (θ + r ) > 0,.
1
x = [ar + a 2 r 2 + 4α (1 + ar ) ]
2

(A10)

Because of (A10), x > ar and then θ > 0 considering x ≡ a(θ + r ) > 0 with a > 0 . Then,the relationship between the optimal growth rate θ, the preference parameter α, the realinterest rate r and the parameter of money demand a is as follows:1a(θ + r ) = [ar + a 2 r 2 + 4α (1 + ar ) ], a 2 (2θ + r ) 2 = a 2 r 2 + 4α (1 + ar ),22a θ (θ + r )α=
1 + ar13(A11)

where (A11) can be interpreted as a solution for the optimal money growth θ, given
other parameters. Alternatively, it is interpreted that the unknown preference parameterα except for the government is expressed by the observed (optimal) money growth θ,
the observed real interest rate r and the estimated money demand parameter a.
■

14

References
Amano, R.A. (1998), “On the Optimal Seigniorage Hypothesis”, Journal of
Macroeconomics 20: 295–308.
Bailey, M. (1956), “The welfare cost of inflationary finance”, Journal of Political
Economy 64: 93–110.
Blanchard, O.J. and Fischer, S. (1989), “Lecture on Macroeconomics”, MIT Press,
Cambridge USA.
Gonzalo, J. (1994), “Five alternative methods of estimating long run equilibrium
relationships”, Journal of Econometrics 60: 203–233.
Hansen, H. and Juselius, K. (1995) CATS in RATS; Cointegration Analysis of Time
Series, ESTIMA.
Ho, T-W. (2003), “Regime-switching properties of the optimal seigniorage hypothesis:
the case of Taiwan”, Applied Economics 35: 485–494.
Honohan, P. (1996), “Does it matter how seigniorage is measured?”, Applied Financial
Economics 6: 293–300.
Johansen, S. and Juselius, K. (1990), “Maximum likelihood estimation and inferenceson cointegration, with applications to the demand for money”, Oxford Bulletin of
Economics and Statistics 52: 169–210.
Johansen, S. and Juselius, K. (1992), “Testing structural hypotheses in a multivariate
cointegration analysis of the PPP and the UIP for UK,” Journal of Econometrics
53: 211–244.
Loviscek, A. (1996), “Seigniorage and the Mexican financial crisis”, The Quarterly
Review of Economics and Finance 36: 55–64.
Mankiw, N.G. (1987), “The optimal collection of seigniorage: theory and evidence”,
Journal of Monetary Economics 20: 327–341.
Marty, A.L. (1976), “A note on the welfare cost of money creation”, Journal of
Monetary Economics 2: 121–124.
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277–301.
Phylaktis, K. and Taylor, M.P. (1993), “Money demand, the Cagan model and theinflation tax: some Latin American evidence”, The Review of Economics andStatistics 75: 32–37.
Sidrauski, M. (1967), “Rational choice and patterns of growth in a monetary economy”,
American Economic Review 57: 534–544.
Tekin-Koru, A. and Özmen, E. (2003), “Budget deficits, money growth and inflation:
the Turkish evidence”, Applied Economics 35: 591–596.
Turner, P. and Benavides, G. (2001), “The demand for money and inflation in Mexico
1980–1999: implications for stability and real seigniorage revenues”, Applied
Economics Letters 8: 775–778.

15

Table 1. Augmented Dickey–Fuller (ADF) test for a unit root in the data

R
HmoneyW
HmoneyC
DmoneyW
DmoneyC

Level τµFirst difference τµLevel ττFirst difference ττ–2.33 (4)0.77 (3)1.58 (1)–0.84 (3)1.50 (6)–3.14 (5)–6.01 (2)
–6.29 (2)–8.07 (2)–5.28 (5)–1.47 (4)–2.61 (3)–2.22 (1)–0.29 (3)–3.25 (6)–4.93 (3)–6.19 (2)–6.72 (2)–8.07 (2)–5.78 (5)Notes: τµ and ττ are the test statistics allowing for constant mean and trend in mean, respectively.
The reported numbers in the columns are the ADF statistics. Numbers in parenthesis after thesestatistics indicate the lag length used. The critical value for sample size 250 at the 0.05
significance level is –2.88 for τµ and –3.43 for ττ.

Table 2. Determination of lag lengths (multivariate LB test and SBIC test)
LB
149.75138.10185.43196.53

R-HmoneyW
R-HmoneyC
R-DmoneyW
R-DmoneyC

d.f.
180
180
172
160

p-value(0.95)(0.99)(0.23)(0.03)

Resulting lag
2
2
4
6

Notes: R-HmoneyW is an error correction model consisting of interest rate R and HmoneyW.
The other labeling is similar to this. The multivariate LB test statistic under the null hypothesisof the uncorrelatedness of residuals has an asymptotic χ2 distribution: see Hansen and Juselius(1995, p. 73). The values in parentheses denote p-values.

Table 3. Tests of cointegration ranks
Rank
Null hypothesis: H0
Alternative: H1
R-HmoneyW
R-HmoneyC
R-DmoneyW
R-DmoneyC
Critical values, 5%

λ-Max
r=0
r=1
r=1
r=2
32.45
5.52
27.71
6.55
29.50
7.79
43.55
7.53
15.67
9.24

Trace
r=0
r=1
r=1
r=2
37.97
5.52
34.26
6.55
37.29
7.79
51.08
7.53
19.96
9.24

Resulting ranks

1
1
1
1

Note: See the notes of Table 2. The r denotes the number of cointegrating vectors. The 5%
critical values of the maximum eigenvalue (λ-max) and the trace statistics are taken from
Osterwald-Lenum (1992, pp. 468).

Table 4. Estimates of money demand function
An optimal money growth % (θ* = 1/a)for seigniorage revenue
3.08%
2.26%
11.90%
9.17%

ln(m) = C − aRt (19)HmoneyW
HmoneyC
DmoneyW
DmoneyCln(m) = 10.784 − 0.325Rt
ln(m) = 11.395 − 0.442Rt
ln(m) = 11.463 − 0.084Rt
ln(m) = 11.615− 0.109Rt

16

Table 5. Test for H0: θ = θ*: optimal money growth for seigniorage revenue
H0: θ = θ*HmoneyW
HmoneyC
DmoneyW
HmoneyC

Av. rates θ A
0.46
0.46
0.23
0.23

3.08%
2.26%
11.90%
9.17%

Std. of θ: σθ
0.94
0.94
0.52
0.52

t-ratio: t = ( θ A –θ*)/σθ
−2.79**−1.91*−22.44**–17.19**Notes: ** and * indicate significance at the 5% and 10% level, respectively. Av. θ A and Std. σθ
are the mean and the standard deviation, respectively.

Table 6. Test for H0: R = R*: optimal money growth for consumer revenueR

H0: R = R*
0%

Std. of R: σR
1.92

Av. rates R A
3.37

t-ratio: t = ( R A − R*)/σR
1.76*

Notes: See notes of Table 5. The zero-truncated normal distribution for R is assumed.

Table 7. Seigniorage revenue aversion α: 1990s and 2000s
HmoneyW
HmoneyC
DmoneyW
DmoneyC

1990s
2000s
1990s
2000s
1990s
2000s
1990s
2000s

a

θA

r = R A −θ A

α in (18)

0.325

0.414
0.554
0.414
0.554
0.265
0.181
0.265
0.181

3.882 = 4.296 – 0.414
1.301 = 1.855 – 0.554
3.882 = 4.296 – 0.414
1.301 = 1.855 – 0.554
4.031 = 4.296 – 0.265
1.674 = 1.855 – 0.181
4.031 = 4.296 – 0.265
1.674 = 1.855 – 0.181

0.083
0.076
0.128
0.127
0.006
0.002
0.010
0.003

0.442
0.084
0.109

Notes: See notes of Table 5. The 1990s and 2000s mean 1990:1–1999:12 and 2000:1–2005:12,
respectively.

17

Figure 1R
R=θ+ rS
r0

m

Figure 2

W =W

S

W = S
W =U
0

U
−S

18

Figure 3

SA

S (θ *)U (θ ∗ ∗ )0U (θ *)U

S = S (θ ∗ ∗ )BFigure 4

A

S

0

U
B

19

Figure 5. Data:
High-powered real money, real M2 + CD and production index in logarithms
12

4.7

ln ( D m o n e y C )11
4.6
ln(DmoneyW)

10lnyln ( y )4.59ln ( H m o n e y C )ln(HmoneyW)

2005M5

2004M9

2004M1

2003M5

2002M9

2002M1

2001M5

2000M9

2000M1

1999M5

1998M9

1998M1

1997M5

1996M9

1996M1

1995M5

1994M9

1994M1

1993M5

1992M9

1992M1

1991M5

1990M9

4.4
1990M18

Note: moneys are measured in 100 millions at right left axis and production in no unit atright axis.
Lending rate and growth rates of high-powered nominal money and nominal
M2+CD (%)
8
High-p ower ed M oney
L ending r a t e

64
M 2+C D
2

-4
-6

20

2004M1

2002M1

2000M1

1998M1

1996M1

1994M1

1992M1-2

1990M10Figure 6

S

A

1990s

2000s

0

U
B

21