Monetary Policy Arithmetic, Superneutrality, and the
Empirical Evidence: A Reconciliation
Maxim Nikitin
Department of Economics
University of Alberta
Edmonton, AB T6G 2H4, Canada
Maxim.nikitin@ualberta.ca
Steven Russell
Department of Economics
Indiana University - Purdue University at Indianapolis
425 University Blvd.
Indianapolis, IN 46202, USA
Email: shrusse@iupui.edu
March 14, 2003
Abstract
Existing "unpleasant monetarist arithmetic" models often generate predictions inconsistent
with empirical evidence. They do not allow for superneutrality of money, and they predict
that a monetary tightening reduces the real interest rate in the long-run, unless the economy
is either on the downward-sloping side of the Laer curve, or is dynamically ineÆcient. This
paper reconciles the monetary policy arithmetic approach with empirical evidence. We show
that, under an alternative money demand assumption that has been widely used in innitehorizon models, money is superneutral, regardless of the level of interest rates or of the dynamic
eÆciency/ineÆciency of the steady state. We also show that under a reduction in the money
supply growth may cause the real interest rate to rise, and output to fall, even though the steady
state is dynamically eÆcient. We derive analytically the conditions under which monetary
tightening impacts on the real interest rate positively and negatively.
KEYWORDS: Monetary Policy Arithmetic, Superneutrality, Government Budget Constraint,
Seigniorage
JEL Classication Numbers: E60, E13, E40
We are grateful to John Duy, Roy Gardner, Juergen von Hagen, Michael Haliassos, Peter Ireland, Hugh LloydEllis, Bennett McCallum, Todd Smith, Christopher Waller, as well as the seminar participants at UofA, Pitt, ZEIUniversity of Bonn, and the New Economics School for their helpful comments on earlier drafts of the paper. All
remaining errors are our own.
1 Introduction
The seminal work of Sargent and Wallace (1981) on \unpleasant monetarist arithmetic" has been
followed up in a stream of literature. This literature analyzes the interaction of scal and monetary
policy in overlapping generations models with currency, government bonds and private loans or
capital. What happens if the government decides to increase its reliance on bonds to nance a
given primary budget decit? Sargent and Wallace (1981) present an example in which a higher
rate of debt creation raises the interest spending of the scal authority, and hence increases, not
reduces, the need for seigniorage. Thus the ination rate is even higher than would have been
necessary had the government chosen at the outset to nance the decit by printing money.
Although the original Sargent and Wallace (1981) paper had an endowment economy and a
xed real interest rate, subsequent contributions extended the analysis to environments with an
endogenously determined real interest rate and a neoclassical production function. The current
literature distiguishes the cases of \Pleasant" and \Unpleasant" monetarist arithmetic (abbreviated
as PMA and UMA, respectivey). In the UMA case there is a positive relationship between the
steady state ination rate, on the one hand, and the real interest rate and the stock of government
bonds, on the other hand. An increase in the stock of bonds raises the real cost of servicing the
government debt. Therefore, to satisfy the government budget constraint, the money seigniorage
and the money growth rate have to rise. At the same time, a larger stock of government debt
crowds out private loans and/or productive capital, and hence it raises the real interest rate.
In the PMA case, there is a negative relationship between the ination rate and the real interest
rate in the steady state. This is possible in two situations. First, if the economy is dynamically
ineÆcient, i.e., if the real output growth rate is higher than the return on non-monetary assets, the
bond seigniorage is positive, and renancing the larger stock of bonds reduces the need for money
seigniorage. Second, if the economy is on the downward-sloping side of the Laer curve, i.e., if
an increase in the money supply growth rate lowers the money seigniorage, the government has to
reduce the rate of money creation to nance a larger debt.
Unfortunately, existing monetarist policy arithmetic models generate predictions inconsistent
with empirical evidence. This evidence suggests that an increase in the steady-state ination
rate has either no impact on capital accumulation and output (superneutrality), or, when the
1
ination rate is low, has a moderate positive impact. However, existing models do not allow
for superneutrality of money. Moreover, they predict that a disination reduces the real interest
rates in the long-run (and hence raises the capital stock and output in models with a neoclassical
production function), unless the economy is on the downward-sloping side of the Laer curve, or
unless it is dynamically ineÆcient.
Bhattacharya, Guzman, and Smith (1998) use a pure-exchange economy model and nd that
UMA always holds on the upward-sloping side of the Laer curve, regardless of whether the economy is dynamically eÆcient. Bhattacharya and Kudoh (2002) assume the neoclassical production
economy, and conrm that UMA necessarily holds if the real interest rate exceeds the output growth
rate. Espinosa-Vega and Russell (1998) generate an opposite (and empirically relevant) PMA result
in a dynamically ineÆcient pure exchange economy. However, the assumption of dynamic ineÆciency is not empirically plausible. In modern economies the return on many types of private sector
liabilities, including diversied portfolios of common stocks, does exceed the average growth rate.
Espinosa-Vega and Russell (2001) extend the earlier Espinosa-Vega and Russell (1998) result.
They obtain the PMA prediction in a neoclassical production economy, which also allows for dynamic eÆciency. They introduce a wedge between the return on government bonds (which is still
lower than the growth rate of the economy), and the return on capital (which may be higher than
the growth rate). However, they do not provide any closed-form results regarding the PMA under
an assumption of dynamic eÆciency.
This paper reconciles the monetary policy arithmetic approach with empirical evidence. We
replace the reserve requirement as a motive for holding money when it is dominated in rate of
return, with the money in the utility function, a specication that has been used in innite-horizon
models. We show that, under this alternative money demand assumption, money is superneutral
if the utility-from-money term is logarithmic. This result holds regardless of the level of interest
rates or of the dynamic eÆciency/ineÆciency of the steady state.
We show that, under a generalized version of this utility function, increases in the money supply
growth rate may cause real interest rate to fall and output to rise (the PMA result), even though
the steady state is dynamically eÆcient, and the economy is on the upward-sloping part of the
Laer curve. We are able to provide analytical conditions for this empirically relevant PMA result.
1
1
See King and Watson (1997), Bullard and Keating (1995), Ahmed and Rogers (1998), Bruno and Easterly (1998),
Crosby and Otto (1999), Rapach (1999). Section 2 of this paper surveys the empirical evidence.
2
Our results are driven by higher sensitivity of money demand to changes in the nominal interest
rate than is the case in OLG models with reserve requirements or random relocation. Similar to
those models, our model implies that a reduction in money seigniorage, following a reduction in
the steady-state ination rate, must be oset by an increase in bond seigniorage. However, a
higher substitutability between money and bonds in portfolios of private agents means that this
can be achieved through adjustment of the stock of bonds rather than by a change in the return
on government bonds. Therefore, a change in the steady-state ination rate can be superneutral,
i.e., it can have no impact on the real interest rate, capital stock, output and consumption.
Our results also contribute to the money and growth literature which studies the impact of a
change in the money supply growth rate on the capital stock and output. We obtain the empirically
plausible superneutrality result in a model with overlapping generations without bequests. Until
now the steady-state superneutrality of money has been found only in models with an innitely
lived representative agent. This result has been reproduced in an alternative setting, such as an
OLG model, only if agents are assumed to care about the utility of their ospring, and so bequeath
part of their wealth to them (Carmichael, 1982). The assumption of intergenerational altruism and
operative bequests eectively converts an OLG framework into a representative agent framework.
The rest of the paper is structured as follows. Section 2 summarizes the empirical evidence on
the long-run eects of a change in ination rate on output, capital accumulation, and real interest
rates. Section 3 describes the setup of the model and the competitive equilibrium. The comparative
statics results are presented in section 4. Section 5 comprises a conclusion for the analysis.
2
2 Long-Run Eects of a Change in Ination Rate: A Survey of
the Empirical Evidence
Bullard and Keating (1995), King and Watson (1997), and Ahmed and Rogers (1998) analyze the
impact of a permanent shock to the money stock, or ination on real output (GDP). Bullard and
Keating (1995) use data covering 58 developed and developing countries. They nd that in 16
countries there had occurred a permanent shock to ination and a permanent shock to the level of
2
The rst order condition with respect to bequests ensures the invariability of the marginal productivity of capital
in the steady state, and hence guarantees that the capital intensity also remains constant.
3
real output. For these countries they run a bivariate VAR, and nd that the long-run response of
the level of output to a permanent ination shock had been positive and statistically signicant for
four countries, negative and statistically signicant for one country, and not statistically signicant
from zero for 53 countries. They also nd that in 9 countries there had been no permanent shock
to output, but a permanent shock to ination had occurred. This is taken as prima facie evidence
of superneutrality. One more nding of Bullard and Keating (1995) is that low-ination countries
respond to ination shocks dierently from high-ination countries. In particular, for low-ination
countries the point estimate of the long-run response is generally positive, while for high-ination
countries it is zero or negative.
King and Watson (1997) estimate a bivariate VAR of money stock and real output using postWWII US data. They analyze the long-run superneutrality proposition across a range of possible
identications of their system in order to study the robustness of the conclusion to diering specications. They nd a wide range of reasonable identifying restrictions under which superneutrality
cannot be rejected. Although they also nd various specications inconsistent with superneutrality, these rejections are marginal and not robust to all lag-length and sample-period specication
changes.
Ahmed and Rogers (1998) estimate a VECM model using cointegrating relationships among
the real output, ination, consumption, investment, and government/GDP ratio, using US data
from 1889 to 1995. For two dierent specications, the estimates indicate that a permanent shock
to ination increases the level of output, consumption, and investment.
Crosby and Otto (1999) analyze the impact of a permanent shock to ination on the capital
stock. They use the long-run identifying restriction that shocks to capital stock do not have
permanent eects on the rate of ination. They construct capital stock series for 64 countries
using post-WWII data and nd that 34 of these countries have both permanent shocks to ination
and to their capital stock. For these countries Crosby and Otto (1999) test superneutrality with
respect to the capital stock using a bivariate VAR. They nd that a permanent ination shock has
no statistically signicant long-run impact on the capital stock for a large majority of countries.
When the superneutrality hypothesis is rejected, they typically nd a positive impact of ination
on the capital stock.
Several studies analyze the long-run impact of the ination shock on the real interest rate. In
theoretical models with a neoclassical production function there is a monotonic negative relationship
4
between the real interest rate and the capital stock (output). Therefore, the negative eect of an
ination shock on the real interest rate can be interpreted as evidence of a positive impact of
the ination shock on output. On the other hand, evidence supporting the Fisher eect (i.e., the
invariance of the real interest rate to ination rate) can be interpreted as evidence of superneutrality.
King and Watson (1997) test for Fisher eect using a bivariate VAR of CPI ination and
the nominal interest rate. They investigate the hypothesis that the long-run elasticity of the
nominal interest rate with respect to a permanent ination shock is unity across a wide variety
of identication schemes. They nd reasonable specications of the VAR model consistent with
the Fisher hypothesis. However, when statistically signicant deviations from the Fisher relation
occur, the nominal interest rate rises by less than the ination rate, i.e., the long-run impact of the
ination shock on the real interest rate is negative.
Rapach (1999) runs a trivariate VAR of the ination rate, the nominal interest rate, and the
output level using the postwar data for 14 OECD countries. He uses the long-run identifying
restrictions following Blanchard and Quah (1989). For all countries, the point estimates indicate
that real interest rates fall in response to a permanent ination shock. Moreover, these eects are
often statistically signicant at conventional signicance levels.
Finally, several studies analyzed the impact of a change in the ination rate on the growth rate
of output. Bullard and Keating (1995) nd that the real output growth rates are stationary in
nearly all countries that experienced permanent shocks to ination. This is interpreted as direct
evidence of superneutrality with respect to output growth rates. Bruno and Easterly (1998) using
the data for nearly all of the member countries of the IMF and the World Bank, also nd that
there is no robust evidence of a growth-ination relationship, if the ination crises observations are
excluded.
3
3 Setup of the Model
Consider an overlapping generations model with two generations and production of a single perishable good. In the rst period agents work, and earn wages. In the second period, agents live o
of their savings. There exist three forms of savings: capital (the consumption good of the current
3
Bruno and Easterly (1999) dene the threshold for an ination crisis as the annual CPI ination at or above 40
percent for two consecutive years.
5
period, if invested, becomes a capital good next period), money, and government bonds.
Agents have a time-separable utility function with standard properties, which also positively
depends on real money balances that agents carry from the rst to the second period of life. The
\money-in-the-utility-function" specication used here can be justied by the \shopping time" or
\banking time" microfoundations models.
There is also a government that nances its primary budget decit (assumed to be exogenous
and xed) by either printing outside money or issuing debt (government-backed bonds), or both.
There is no population growth.
Production is described by the production function yt = Akt , where kt is per capita (per young
agent) capital stock, and yt is per capita output. Capital depreciates fully every period.
The nominal money stock evolves according to:
4
Mt+1 = t Mt ;
where Mt is the nominal money stock per capita (per young agent) carried over from period t 1
to period t, and t is the gross growth rate of nominal money supply in period t. We dene the
real money supply mt as PMt t1 , where Pt is the price level at the end of period t 1. Hence the
law of motion for the real money supply is:
5
1
mt+1 = t
mt
;
t
where t is the gross ination factor between periods t 1 and t, and t = PPt t 1 .
The government primary budget decit measured in per capita (per young agent) terms coincides with the government consumption, gt . The budget decit is nanced by seigniorage and
selling bonds:
gt = (t
1) m t + bt
t
+1
bt Rt ;
(1)
where bt is the real stock of one-period bonds at the beginning of the period t, and Rt is the
real gross return on these bonds (by arbitrage it is identically equal to the return on capital).
4
The lower real money balances, the more time agents must spend on shopping and banking (trips to a bank to
replenish cash balances), and the less time they have for leisure.
McCallum and Goodfriend (1987) discuss the \shopping-time" justication for incorporating money balances in the
utility function. Croushore (1993) presents mathematical conditions for the equivalence of models with consumption
and leisure in the utility function and models with consumption and money.
5
This is done for the sake of notational consistency, so that all the stock variables, the real money supply mt , the
capital stock kt , and the real government bonds bt , be indexed by the following period.
6
Because attention is restricted to the steady state in this and the following section, we drop the
time subscripts in what follows.
Agents maximize their utility function subject to their respective budget constraints in both
periods of life:
C C m m
U=
+
+
Æ
(2)
1 1 1 1
1
1
2
1
m
subject to
C1 = w m k b
m
C2 = + R(k + b)
(3)
(4)
where m denotes the real money balances that the young agents accumulate in their rst period of
life.
The gross rate of return on at money is:
1
Rm =
6
In a steady state Rm = .
Agents maximize their utility with respect to real money balances, m, holdings of capital, k, and
bond holdings, b. However, they do not have any preferences between bonds and capital, as long as
expected returns are the same. Therefore, they actually optimize over two variables: total holdings
of the capital market assets, k + b, and m. Optimization results in the rst-order conditions:
1 + R 1 = 0
(5)
1
C1
and
Combining (5) and (6) yields:
C2
1 + Rm 1 + Æ 1 = 0
mm
C
C
1
2
R = Rm +
(6)
ÆC2
mm
In equilibrium there always exists a positive wedge between rates of return on money and capital:
because money enters the utility function directly, agents are willing to keep positive money balances
when the rate of return on money is lower than on capital and bonds.
6
This is a standard way to incorporate money balances in the utility function in an OLG model. See McCallum
(1983) and McCallum (1987).
7
Competitive equilibrium
The competitive steady state equilibrium is determined by the following set of equations:
y = C 1 + C2 + g + k
(7)
(8)
(9)
(10)
(11)
(12)
(13)
y = Ak
R = Ak
1
C2 = (R)1= C1
ÆC R = Rm + 2m
m
C1 = Ak kR k b m
g = (1 Rm )m + b(1 R)
Equation (7) is the goods market clearing condition as well as the standard GDP identity.
Equations (8) and (9) dene the values of output and the marginal return to capital, respectively.
Equations (10) and (11) are the rst-order conditions of the agents' intertemporal optimization.
Equation (12) is the budget constraint of a young agent. Finally, equation (13) is the government
budget constraint.
The equilibrium conditions (7) - (13) determine the seven endogenous variables: y; C ; C ; k; m,
R; b. Rm and g are exogenous variables.
The analysis of the existence of equilibrium, i.e., the existence of a solution to the system (7)(13), is fairly complicated. Appendix 1 derives the existence conditions for the case = m = 1. By
continuity, for any set of the model parameters, such that the solution exists for the case = m = 1,
there is a ball around the point = m = 1, such the solution exists for any combination of and
m inside this ball.
1
2
4 Comparative Statics
General comparative statics analysis of the system (7)-(13) is very complicated. To analyze the
impact of a change in the steady state money supply growth rate on the capital stock and output,
we proceed in three steps. First, we dene the government revenue function
g(R; Rm ) = (1 Rm )m(R; Rm ) + (1 R)b(R; Rm )
8
and replace R with g as an endogenous variable in the system (7)-(13). In other words, we redene
the system, to study how g changes following exogenous changes in R and Rm. Then, we investigate
the relationship between Rm and g holding R constant. Second, we analyze the relationship between
R and g holding Rm constant. Finally, we study how R, the return on the capital market assets,
must change following a change in Rm to keep the government revenue constant.
Lemma 1.
@g
< 0; if and only if
@Rm
@g
> 0; if and only if
@Rm
@g
= 0; if and only if
@Rm
m > 1
m < 1
m = 1
If R does not change, the values of the capital stock and output do not change either.
Totally dierentiating system (7)-(13) under this assumption yields:
Proof:
dC1 + dC2 + dg = 0
dC2 = (R)1= dC1
dRm
m
=
dm
dC
m
R R
m
C2 2
dC1 = db dm
dg = dm mdRm Rm dm + db(1 R)
(14)
(15)
(16)
(17)
(18)
Successive elimination of dC ; dC ; dm, and db yields:
(19)
(1 m)dRm = dg 1 ++ R mm + C 1 + (R Rm ) ;
where = (R) = . The expression in square brackets is always positive. Hence the sign of @R@gm is
the same as the sign of 1 m . Therefore, @R@gm is greater (less) than zero if and only if m is less
(greater) than 1. Q.E.D.
1
2
2
1
Lemma 2.
@g
< 0; if
@R
= m 1; b > 0; R > 1:
Proof: See Appendix.
9
Lemma 3. If
@g
@R
< 0, then
@k
@Rm
> 0 for m > 1;
@k
@Rm
< 0 for m < 1;
@k
@Rm
= 0 for m = 1.
The proof follows immediately from Lemma 1. Totally dierentiating the denition of the
government revenue function, one obtains:
Proof:
dg =
@g
@g
dRm + dR
@Rm
@R
Therefore, if the government revenue remains constant, then
@R
=
@Rm
@g
@Rm
@g
@R
Given that the production function is Cobb-Douglas, and
1
@k sign @R
m = sign
By Lemma 2,
Therefore, by Lemma 1,
A 1
k=
R
;
" @g #
@Rm
@g
@R
@k
@Rm
@k @g sign @Rm = sign @Rm
> 0 for m > 1; @R@km < 0 for m < 1;
@k
@Rm
= 0 for m = 1. Q.E.D.
Proposition 1 below states the condition for superneutrality of money.
@km = 0 if and only if m = 1.
P roposition 1. @R
Proposition 1 imposes no restrictions on , the coeÆcient of relative risk aversion of the
utility-from-consumption terms.
Remark:
Proof:
Taking into account (10), equation (12) can be written as:
C2 (R)
1
= Ak kR k b m
(20)
Analysis of the competitive equilibrium proceeds in the following way. We search for conditions
under which the system (7) - (11), (13), (20) can be decomposed into the \real block" (equations
(7) - (10)), and the \monetary block" [equations (11), (13), (20)], so that a change in Rm aects
only the monetary block variables, but not the real block ones (especially, k; y; and R), and hence
superneutrality holds.
10
The monetary block can be considered a system of three equations with three unknowns: C ; m,
and b, which are functions of k; R and the exogenous parameters.
It is easy to see that among the three endogenous variables of the monetary block only C is
present in (7) - (10). Therefore as long as C is not aected by a change in Rm within the monetary
@C 2
block, i.e.
@Rm = 0, then superneutrality holds.
Total dierentiating the monetary block equations yields:
2
2
2
dRm
= mm dm C dC2
R Rm
2
1
(R) dC1 = db dm
0 = dm mdRm Rmdm + db(1 R)
(21)
(22)
(23)
Successful elimination of db and dm yields:
(R 1)(R)
As (R 1)(R)
m
1
m
(R Rm ) + C
dC2 = (m 1)dRm
m
2
m
1
+ RC2Rm > 0,
(
)
@C2
@Rm
(24)
= 0 if and only if m = 1. Q.E.D.
Proposition 2 below describes the eects of a change in the money supply growth rate in cases the
superneutrality of money does not hold. Although we have to make certain additional assumptions,
these assumptions are empirically plausible. In particular, we have to assume the positive stock of
the government debt and dynamic eÆciency, R > 1.
@km < 0,
P roposition 2. Suppose m = . There exists > 1, such that for any < 1, @R
@km > 0, as long as at the initial steady state R > 1 and b > 0.
and for any (1; ), @R
The proof follows directly from Lemmas 2 and 3. If conditions of Lemma 2 are satised, i.e.,
@g < 0 for (1; ). Therefore,
if b > 0 and R > 1, then by continuity, there exists such that @R
m
by Lemma 3, @R@km < 0 for < 1, and @R@km > 0 for > 1. Q.E.D.
Proof:
Discussion of the Results
Earlier studies of monetary policy arithmetic have used money demand assumptions - reserve
requirements or random relocation - under which money demand is insensitive, or not very sensitive,
to changes in ination rate. Therefore in these models an increase in Rm, the return on money,
11
reduces government revenue from money creation, (1 Rm)m. This reduction must be oset by
higher revenue from bond seigniorage, b(1 R). If the economy is dynamically eÆcient, i.e., if the
return on capital market assets, including government bonds, is greater than unity, then the bond
seigniorage is negative, and the government has to reduce the loss. Low substitutability between
money and bonds in portfolios of private agents means that the real rate of return on government
bonds has to fall to reduce the loss. Hence the return on private assets, including capital, falls as
well, and the steady state capital stock rises. This result is inconsistent with the empirical evidence.
The model developed in this paper overcomes the limitation of the ination insensitive money
demand by utilizing the money-in-the-utility-function motive for holding money dominated in the
rate of return. Furthermore, varying the coeÆcient of relative risk aversion of the utility-frommoney term, m, allows us to study how the sensitivity of money demand aects the direction of
the impact of monetary tightening on the capital stock, real interest rate, and output.
If m is greater than unity, the demand for real money balances rises slowly in response to a
disination, and the \unpleasant monetarist arithmetic" result holds. If m is smaller than unity,
the demand for money rises signicantly in response to a disination, and demand for bonds falls.
Hence, the government can reduce the stock of bonds, even though the return on bonds increases.
This is the \pleasant monetarist arithmetic" result. Finally, if m = 1, the reduction in the supply
of bonds exactly osets the reduction in money seigniorage, so that the real interest rate, capital
stock, and output are not aected. In this case money is superneutral.
5 Conclusions
The paper develops an overlapping-generations model with neoclassical production that takes the
interaction between scal and monetary policy seriously, and accounts for several stylized facts:
a permanent monetary tightening either reduces steady-state levels of output and capital stock,
or leaves them unchanged; the economy is dynamically eÆcient; the economy is on the left, or
upward-sloping, part of the Laer curve. We are able to derive analytically the conditions for the
long-run superneutrality of money, as well as for the \pleasant monetarist arithmetic" result, or
the positive relationship between the return on money and the real interest rate.
Several important issues are left for future research. First, this paper focuses on the steady
states. It is important to analyze the transitional dynamics as well, i.e., to study how the economy
12
moves from one steady state to another following a permanent monetary tightening. Second, the
model does not allow for economic growth in the steady state. A venue for future research is to
incorporate explicitly exogenous technological progress and economic growth. Third, in this model
productive capital and government bonds are perfect substitutes in portfolios of private agents.
The model will be more realistic if there is a wedge between the return on capital and the return
on government bonds, such that R > n > r, where R is the gross real return on capital, r is the
gross real return on government debt, and n is the gross growth rate of real output.
13
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King R. and M. Watson, 1997, Testing Long-Run Neutrality, Federal Reserve Bank of Richmond
Economic Quarterly, 69-101.
14
McCallum B.T., 1983, The Role of Overlapping-Generations Models in Monetary Economics,
Carnegie-Rochester Conference Series on Public Policy 8, 9-44.
McCallum B.T., 1987, The Optimal Ination Rate in an Overlapping-Generations Economy with
Land, in: W.A. Barnett and K.J. Singleton, eds. New Approaches in Monetary Economics.
Cambridge: Cambridge University Press.
McCallum B.T. and M.S. Goodfriend, 1987, Demand for Money: Theoretical Studies, In: J.
Eatwell. M. Milgate, P. Newman, eds., New Palgrave. A Dictionary of Economics, The
McMillan Press Ltd, Vol. 1, 775-781.
Rapach D., 1999, International Evidence on the Long-Term Superneutrality of Money, Working
Paper, Trinity College.
Sargent T. and N. Wallace, 1981, Some Unpleasant Monetarist Arithmetic, Federal Reserve Bank
of Minneapolis Quarterly Review, Fall, 1-17.
15
Appendix
Existence of Equilibrium in Case of the Logarithmic Preferences
This section derives the existence conditions for a particular case of the logarithmic preferences:
= m = 1. In other words, the utility function is:
U = ln(C1 ) + ln(C2 ) + Æ ln(m)
In this case, the steady-state equilibrium conditions (7)-(13) become:
y = C1 + C2 + g + k
(25)
(26)
(27)
(28)
(29)
(30)
(31)
y = Ak
R = Ak
C2 = RC1
R = Rm +
C1 = Ak
1
ÆC2
m
kR k b m
g = (1 Rm )m + b(1 R)
Proposition A1 states conditions for the existence of the steady- state equilibrium described by
the system (25)-(31).
The steady-state equilibrium described by the system (25)-(31) exists if
Proposition A1:
and only if:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
L(k0) R(k0)
2( + + Æ)2 4(1 + + Æ)(1 )(2 1) 0
k0 < k~; or L(k~) > R(k~)
1 +Æ)Ak , R(k) =
, L(k) = (+1+
where k~ = RA
m
+Æ
2
6
6
6
4
k0
k0
p
(++Æ)+ 2 (++Æ)2+4(1++Æ)(1
=
2(1 ) (1 2)A
h (++Æ )A i2
= 2(1++Æ)
16
)A2k02 1 + g + k, and
1+ +Æ
)(1 2) 1=( 1)
(1
for 6= 0:5
for = 0:5
Proof:
Successively eliminating C ; C ; R; m; b and y from the system (25)-(31), one obtains:
1
2
( + + Æ)A k = (1 )A k + g + k
(32)
1+ +Æ
1++Æ
Denote the left-hand side of equation (32) L(k), and the right-hand side R(k). Their derivatives
are: L0(k) = Æ ÆA k and R0(k) = A 2 Æ k + 1. Equilibrium exists if and only
if the equation R(k) = L(k) has a nonnegative solution. Three cases when < 0:5, > 0:5, and
= 0:5 are considered separately.
Case 1. < 0:5. In this case 2 1 < 0 and the function R(k) approaches +1, when k
approaches zero. Its derivative R0(k) is monotonically increasing. It is negative for small values of
k, and positive for large values of k. R(k) > L(k) for small values of k, but the dierence is falling.
It attains the minimum at k , the root of the equation
2
( + + )
1+ +
(1
1
)
(2
1+ +
1)
2
2
1
2
0
L0 (k) = R0 (k)
(33)
and grows back afterwards. This case is depicted on gure 1.
Insert gure 1 here.
Equilibrium exists if and only if
L(k0 ) R(k0 )
(34)
Equation (33) is a quadratic equation in k . The only positive solution is:
p
( + + Æ) + ( + + Æ) + 4(1 + + Æ)(1 ) (1 2)
k =
(35)
2(1 ) (1 2)A
Case 2. > 0:5. In this case 2 1 > 0. The function R(k) is monotonically increasing, but
its derivative is not. Equation (33) has two positive roots:
1
0
( + + Æ)
2
p
2 ( + + Æ)2 4(1 + + Æ)(1 ) (2 1)
2(1 ) (2 1)A
k
1
=
k
1
2
+ + Æ)2 4(1 + + Æ)(1 ) (2 1)
= ( + + Æ) + (2(1
) (2 1)A
2
and
2
1
1
p
(36)
(37)
Needless to say, these roots are real and positive only if the discriminant of the quadratic
equation is nonnengative, or
2 ( + + Æ)2
4(1 + + Æ)(1 ) (2 1) 0
17
It is straightforward to verify that the derivative of R(k) is greater than the derivative of L(k)
for k < k and k > k , and smaller for k < k < k . R(k) > L(k) for small values of k, and the
dierence rises until k = k . Then R(k) L(k) falls until k = k , and rises again afterwards to
become innitely large. Therefore k is the point at which R(k) L(k) is the most likely to be
negative.
Figure 2 depicts this case.
1
2
1
2
1
2
2
Insert gure 2 here.
The equilibrium exists if and only if:
R(k2 ) L(k2 )
(38)
It is easy to see that k = k .
2
0
Case 3. = 0:5. R(k) = A
Æ + g + k , and R (k ) = 1, i.e., R(k ) is a straight line.
R(k) > L(k) for small values of k, and the dierence falls until k = k , where k is the solution of:
L0 (k) = 1, or
2
0
(1
)
1+ +
0
0
( + + Æ)A (39)
2(1 + + Æ)
and rises for k > k . Hence, the solution of (32) exists if and only if L(k ) R(k ).
The existence of solution of the equation L(k) = R(k) is a necessary but not suÆcient condition
for the existence of equilibrium. Another condition is that R(k) > Rm , where k solves (32). Dene
k0 =
2
0
0
k~ =
A 1
Rm
0
k~ is dened in such a way that Rk (k~) = Rm . Hence the condition Rk > Rm is equivalent to:
k < k~
(40)
(41)
The last inequality is satised, if either k < k~, or k k~, but L(k~) R(k~).
Therefore the following is the necessary and suÆcient condition for the existence of a steadystate equilibrium described by the system (25)-(31):
0
8
>
<
2 ( + + Æ)2
>
:
0
L(k ) R(k )
4(1 + + Æ)(1 ) (2 1) 0
k < k~; or L(k~) > R(k~)
0
0
18
0
(42)
where k~ =
2
6
6 k0
4
A
Rm
=
k =
0
1 and
p
(+ +Æ)+ 2 (+ +Æ)2 +4(1+ +Æ)(1 ) (1
2(1 ) (1 2)A
h (++Æ)A i2
2(1+ +Æ )
19
2 )
1=(
1)
for 6= 0:5
for = 0:5
(43)
Figure 1
R (k )
6
L(k)
k0
6
Figure 2
k
R (k )
L(k)
k1
k
k2
20
Proof of Lemma 2
Let (R ) = and Z 1
RÆ
R Rm
1=
. The maximization of the utility function:
U (C1 ; C2 ; m) =
C11
C
m
+
+
Æ
1 1 1 1
2
1
subject to the combined constraint of the household:
C1 +
C2
=w m 1
R
Rm
R
yields the following household demand functions:
C1 = w
R
:
R + + (R R m ) Z
m = Z C1 = w
and we have
RZ
:
R + + (R R m ) Z
C1 + m = (1 + Z ) C1 = w
The residual demand for capital is
R (1 + Z )
:
R + + (R Rm ) Z
b = w C1 m k
m
= w R + R++(R +R(R) Z RmR) Z(1 + Z )
m
= w R + + (RR ZRm) Z k :
k
The equilibrium seigniorage revenue function is
g(R; Rm ) = (1 Rm ) m + (1 R) b
Rm Z
k
= (1 Rm ) w R + +R(RZ Rm) Z + (1 R) w R + + (R
Rm ) Z
= R + + (wR Rm) Z [(1 Rm) R Z + (1 R) ( Rm Z )] (1 R) k
m
= (R 1) k + w ((RR RRm))ZZ +((RR +1)) :
Now
@g
@m
@b
=
(1
Rm )
+
(1
R)
@R
@R
@R
21
b:
@g
@b
Again, if we assume b > 0 then (1 Rm) @m
@R + (1 R) @R < 0 is suÆcient for @R < 0. Now,
(R Rm) Z (R 1) +
@g
=
k + (R 1) k0 + w0
@R
(R Rm ) Z + (R + )
[Z + (R Rm ) Z 0 (R 1) 0 ] [(R Rm) Z + (R + )]
[Z + (R Rm ) Z 0 + 1 + 0] [(R Rm) Z (R 1) ]
w
[(R Rm ) Z + (R + )]
In addition, we have
Rm Z
b = w
(R R m ) Z + + R k
(
)
[(
R Rm ) Z + + R] [ Rm Z ]
= w
k:
[(R Rm) Z + + R]
Some tedious arithmetic then gives us
2
2
@b
+ (1 R)) @R
(1 Rm) @m
@R m
= (R 1) k0 + w0 ((RR RRm))ZZ +((RR +1))
m
m
0
0
m
+w [(R R ) Z + (R 1)] [( mR Z ) R ] + R Z (R R ) (1 + ) :
[(R R ) Z + (R + )]
So we need
)
(
(
R m Z ) R 0
0
0
(R 1) k w (R Rm ) Z + (R + ) + w
[(R Rm) Z + (R + )]
(R R m ) Z
(R Rm) fZ [( Rm Z ) R 0 ] + R Z 0 (1 + )g :
< w0
w
(R Rm ) Z + (R + )
[(R Rm) Z + (R + )]
We can rewrite this condition as
(R 1) k0 w0 (R Rm) Z + (R + )
(R R m ) Z
[(R 1) + (R Rm) Z ] [( Rm Z ) R 0 ] + (R Rm ) [R Z 0 (1 + )] :
< w0
w
(R Rm) Z + (R + )
[(R Rm) Z + (R + )]
If we could show that the second term on the RHS of the above inequality was always positive,
then we would need
k0
1 >
=
:
0
m
w R (1 ) (R R ) Z + (R + )
2
2
2
2
Note also that
(
R Rm ) Z (R 1) g = (R 1) k + w
(R Rm) Z + (R + )
m
= (R 1) k w (R Rm ) Z + (R + ) + w (R (RRm) ZR+ )(ZR + )
22
Dene = (R Rm) Z + (R + ). Then we have
w (R R m ) Z
w
+
>0
g > 0as(R 1) 1
k k
and
@g
w0 w 0 (R R m ) Z
< 0as(R 1) 1
+
0
@R
k k0
m
w [(R 1) + (R R ) Z ] [( Rm Z ) R 0 ] + (R Rm ) [R Z 0 (1 + )]
:
>
k0
2
If we were sure that expression on the RHS of the last inequality was negative, then it would
be suÆcient for us to show that
(R 1) 1
implies
(R 1) 1
We can rewrite the rst condition as
w w (R R m ) Z
+k
k 0
w 0 w 0 (R R m ) Z
+ k0
> 0:
k0 (R 1) 1 wk [ (R Rm ) Z ] 0 ;
so we know that
1 w [ (R Rm) Z ] R 1
k
We can rewrite the second condition as
0
(R 1) 1 wk0 [ (R Rm) Z ] > 0 :
Now
w0
= R (1 )
k0
and
w
=
k
So
(1 )
1
R
1 1
R
= R (1 ) :
w0
w
=
:
0
k
k
23
Thus,
1 w0 [ (R Rm) Z ] = 1 w [ (R Rm) Z ]
k0
k
and it follows that g 0 implies
w 0 w 0 (R R m ) Z
(R 1) 1 k0 + k0
> 0:
As we have seen, this would be suÆcient for @g=@R < 0 if we could show that
[(R 1) + (R Rm ) Z ] [( Rm Z ) R 0 ] + (R Rm ) [R Z 0 (1 + )] < 0 :
Since Z 0 < 0, a suÆcient condition for this latter term to be negative is
2
R m Z < R 0
which is
=
(R ) = (1 1 ) < Rm R R ÆRm
:
1
1
If 1 then this condition always holds. Q.E.D.
24