Theoretical Economics Letters, 2011, 1, 91-94
doi:10.4236/tel.[phone] Published Online November 2011 (http://www.SciRP.org/journal/tel)
A Comment on Reis*
Kenji Miyazaki
Faculty of Economics, Hosei University, Tokyo, Japan
Received July 17, 201 1; revised August 31, 2011; accepted September 8, 20 1 1
Abstract
This note gives a counterexample on Reis [1]. Using a certain family of utility functions, this note not only
gives a sharper representation than that of Reis but also demonstrates that interest rate inelastic money de-
mand does not lead to superneutrality. This implies that superneutrality does not exist when uncerinty is in-
troduced.
Keywords: Monetary Policy, Superneutrality, Nominal Interest Rate Policy, Perfect Complementary between
Consumption and Money
1. Introduction
Reis [1] characterized the dynamics of the money-in-the-
tility model (Sidrauski, [2]) by using the money demand
function to explain the mechanism in a very intuitive
manner. One of his main conclusions is that when as-
ming that the government can control nominal interest
rates by setting any growth rate of money supply, mone-
try policy does not affect any level of consumption and
capital stock as long as either money demand is inelastic
with respect to nominal interest or money and consump-
tion are separable in the utility function. Subsequently,
Lioui and Poncet [3 ] attached uncertainty with Reis’ fra-
mework to demonstrate that superneutrality is valid only
in the case of an interest rate inelastic money demand.
However, both studies do not pursue a sufficient investi-
gation on the relationship between the money demand
function and the utility fu nction.
This note gives a counterexample for their statements.
That is, we show that within a certain family of utility
functions, interest rate inelastic money demand does not
lead to superneutrality. An intuitive explanation is as
follows. A nominal interest monetary policy affects real
variables through the product of the interest rate elasticity
of money demand and the elasticity of the marginal utility
of consumption with respect to money. When conmption
and money are perfectly complementary, the former elas-
ticity is zero but the latter elasticity takes infinity. When
the product of both elasticitie s converges to a finite valu e,
such a policy is still effective.
2. S-Sidrauski Economy
In order to prepare a counterexample, this section briefly
reviews a Reis-Sidrauski economy and reconsiders the
assumptions on the utility fun c tion of Reis [1].
In the economy, , , and , respect-
tively, denote consumption, capital stock, and real bal-
ances or just money. Technology is characterized by a
constant parameter
0
t
c
0
0
t
k0
t
m


()fk of depreciation rate and a production function t with , 00
k
f0
kk
f

,
(0) 0f

, 0 limkk f

, and  . Represen-
tative agents are infin ity lived with perfect foresight, and
their preferences are characterized by a constant para-
meter
lim 0
kk
f
0

 of the rate of time preference and a utility
function ()
tt
uc m

.
A set of assumptions imposed on is discussed later.
In equilibrium, the representative agent maximizes their
lifetime utility to choose t and t, the markets are
clear, and the government chooses nominal interest rates
t
u
c m
()
tkt
Rfk


, where t

denotes inflation rates, by
controlling an appropriate rate of money growth.
The equilibrium dynamics system is characterized by
the money demand function (cR)

, defined by
()(
mc
Ruc uc)



, which results from the necessary
condition for the maximization problem of the representa-
tive agents. Using

, we descri be the dy namics system 1 as
*The author is grateful for a grant-in-aid from the Ministry of Educa-
tion and Science, the Government of Japan (21530277).
1In the conventional monetary policy with a constant rate of money
growth

, we should add //
k
R
Rccf R   
to the two equa-
tions in order to describe the system.
k
cR
f
cR
kf kc




 




K. MIYAZAKI
92
where
(( ))(( )),
(())(()),
()(),
()(),
cc c
cm c
R
c
cuccRuccR
mucc Rucc R
RcR cR
ccR cR



 
 
 
 

 
 
respectively, represent the inverse of the intertemporal
elasticity of substitution, the elasticity of the marginal
utility of consumption with respect to real money bal-
ances, the interest rate elasticity of money demand, and
the consumption elasticity of money demand.
Reis [1], in his proposition 2, stated that money is su-
perneutral when


0
is equal to zero. Following the
proposition, Reis stated that such superneutrality attains
either if money and consumption are separable in the
utility function (

) or if money demand is inelastic
with respect to nominal interest (0

0). In this no te, we
give a counterexample satisfyin g

 but 0



.
Before providing the example, we discuss a set of as-
sumptions regarding the utility function. Reis [1] as-
sumed , , , , ,
and . When we assume , then
0
c
u
0
cm 0
cc
u0
m
uu0
mm
u
0
mcc nncm
uu u
u0R
2
u
mc , implying that the government should set
zero nominal interest rates. In addition, when we assume
cc mmcm , then, as shown later, we cannot exclude the
possibility of
uu
uu 0

. The assumption cm is a little
bit restrictive because this assumption excludes the
case of
0u
1

 in the famous CRRA form of
11
()m(c)m

(

1)uc


 , where 01

 is a con-
stant parameter.
Instead of the above assumptions on the utility func-
tion, we propose the following assumption: ,
, , , ,
cm , and
0
c
u
0
cc
u
m
uu 0
m
u0
mm c
u u0
mm
u

20
cc nncm
uu u
0
ccm ccm
u uuu

 for all
and . The first four assumptions indicate that
is strictly increasing and strictly concave with respect to
and . The last two assumptions arise from
0cu0
m
m
c

()uu c0
mc and () 0 mc
uu m. These assump-
tions are the same as those of Fischer [4]. Using the total
differential form:
d{()}{()}d
mc mc
Ruucdcuum  m,
we obtain
2
()
c
Rmmccmm mcR
u
uu uu



 (1)
()
cc mc cm
c
mmccmm mcR
uu uu
uu uu





and
2
()
cc mmcm
mmccmm mcR
uu u
cuu uu







Therefore, if the above assumptions are satisfied, then
R


,c

, and

are all nonnegative. When cm mmm c uuu u

and ccmcc
um uuu

are finite, then
R

,c

, and

are
all positive.
From equation (1), the interest rate elasticity of money
demand R
R



 might takes zero only when
cm mmm c
uuu u

takes infinity, This would happen when
cm or ucm c
mu u


takes infinity. This makes us
conjecture that, even when 0

, the product of

and

is not necessari ly zero.
3. Counter Example
Because we cannot prove the conjecture in the above
general class of utility functions, we set a somewhat
restrictive class to give a counterexample. Let
()(()ucmwcz)



cw w, where . When 0zmc





m
is constant, this is exactly the class of utility
functions Lucas [5] proposed. In order for u to be
strictly increasing and strictly concave with respect to
and , respectively, we assume that and c w

are
strictly increasing and strictly concave, respectively, and
01z




 for all . 0z
Under this class, the money demand function is deter-
mined by
()
() ()
z
Rzzz





 (2)
The right-hand side of the abov e equation is pos itive and
strictly decreasing for all ,2 and, accordingly, there
exists an inverse function 0z
z
()R


()R. Thus, the money demand function mc


is well-defined. The elas-
ticities of the money demand function with respect to
consumption and interest rates are, respectively, unity
and
()
()(){ ()()}
( )()()
z
R
RRzzzz
Rzzz




 


  
)
The last equality is established by using Equatio n (1 ) and
()(()ucmwcz


.
The dynamic is described as the same in the previous
section and the coefficients are expressed in a simpler
way. With some algebraic operations3, we can get
()
z
R
cww



 

and
()
()
(1) ()
z
R
zz
z







 (3)
2In fact, 2
d() ()()
0
d()()() ()
zzz
zzzz{zzz}

 




3See Appendix.

K. MIYAZAKI
93
Equation (3) indicates that the elasticity of the shadow
price c with respect to money is represented much
more clearly than that of Reis [1]. That is, the elasticity
u

is determined by

,

, and the relative slope of

.
When 1

  or 1

 , then the interest elasticity of
money demand is smaller than the elasticity of the in-
tertemporal substitutio n. In this case, the shadow price of
capital is increasing in money. When 1

 , then
or the utility function is separable.
0
cm
u
Because (1 )z




 and 01z




, we
can show 0

 but 0

 
0 within our family of util- ity function. Even if

,

is growing much faster,
and, accordingly,

 converges to z


. Only when
the utility function is separable does

 take the value
of zero.
Finally, we present a parametric example. The utility
function is described as
(1 ) 11
11
[(1)]{( )}
() 1
cm cz
ucm
 

1

 

 




 

where 01

, 0

, and 0

 are constant pa-
rameters and 11 () [1]zz




 
m
 for .
Notice that when 0zmc 0min[ ]zc

 and that
1
zcm

 
 when 1

. Consumption and real balances
are perfect complements when 0

. The case of 0


corresponds the case of a cash-in-advance economy, in
which money is needed for purchasi ng consumption goods
and the cash-in-advance constraint is always binding4.
In this case, the elasticity of intertemporal substitution
and the interest rate elasticity are respectively determined
by the constant parameters 1

and

, and

 is
represented as a function only of , or R

1
11
(1 )
R
R















Clearly, (1 )RR

 when 0

 and
(1 )

 

 when 1


. when 1

 ,

 takes
zero.
4. Concluding Remarks
In summary, using a larger set of utility functions than
that of Lucas [5], we not only give a sharper representa-
tion than that of Reis [1] but also give a counterexample.
When consumption and real balances are perfectly com-
plement, then the interest rate elasticity of money de-
mand is zero but a nominal interest policy is not su-
perneutral. Only in the case of a separable utility func-
tion does superneutrality survive. This discussion as-
sumes that consumers have perfect foresight and no un-
certainty exists. When uncertainty is introduced, follow-
ing Lioui and Poncet [3], separability does not assure
superneutrality. Therefore, no superneutrality exists with
our family of utility functions.
5. References
[1] R. Reis, “The Analytics of Monetary Non-Neutrality in
the Sidrauski Model,” Economics Letters, Vol. 94, No. 1,
2007, pp. 129-135. doi:10.1016/j.econlet.2006.08.017
[2] M. Sidrauski, “The Rational Choice and Patterns of Gro-
wth in a Monetary Economy,” American Economic Re-
view, Vol. 57, No. 2, 1967, pp. 534-544.
[3] A. Lioui and P. Poncet, “Monetary Non-Neutrality in the
Sidrauski Model under Uncertainty,” Economics Letters,
Vol. 100, No. 1, 2008, pp. 22-26.
doi:10.1016/j.econlet.2007.10.023
[4] S. Fischer, “Capital Accumulation on the Transition Path
in a Monetary Optimizing Model,” Econometrica, Vol.
47, No. 6, 1979, pp. 1433-1439.
doi:10.2307/1914010
[5] R. E. Lucas Jr., “Inflation and Welfare,” Econometrica,
Vol. 68, No. 2, 2000, pp. 247–274.
doi:10.1111/1468-0262.00109
4The constraint is binding when the government sets the nomi-mc
nal interest rate to be
p
ositive.

K. MIYAZAKI
94

Appendix
Consider where ()()ucm wy()ymcc

. The de-
rivatives of are described as follows: u
2
23
{( )( )}()
()()
{()()} ()
()()()
c
m
cc
umcmcmcwy
umcwy
umcmcmcw
mc mcwy
y










 


2
2
{( )}()( )(1)()
(){()()}(
()()()
mm
cm
umcwymccw
umcmcmcmcw
mcmcw y



 


 
 

)
y
y

m
The money demand function is derived from Equation
(2). The total differential form is described as
, where d{( )}d{( )}d
mcmc
Ruuccuum   
2
22
() ()()()
{( )( )( )}
mcccmmcc
c
uuuu uumcmc mc
cumcmc




 

22
() (1 ) ()()
{( )( )()}
mccmm mcm
c
uu uu uucmcmc
mumcmcm




c


 
Using {( )}{( )}
cmcmc
uu cuu mmc


   and
1{ ()}
mc
uu m

, we obtain:
{() }
(){()()()
()()()
Rmc
Rmuu m
mcmcmcmc
mc mcmc

 

 


 
}
{( )}{( )}1
mc mc
cuucmuum



Because of 1


, ()()
()
cc cm
cc
cu mucmcwy
uu wy




 
Because of
mc
22
() ()()
()() ()
cm
c
mu mcmcm cmc
umcmcmc



mc



 
w
e obtain Equa t i on (3 ).