GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.40

Velocity and Monetary Expansion
in a Growing Economy with Interest-Rate Control

Seiya Fujisaki

March 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Velocity and Monetary Expansion in a Growing
Economy with Interest-Rate Control ∗Seiya Fujisaki†
March, 2009

Abstract
We analyze the income velocity of money in an endogenous growth modelwith an interest-rate control rule and a cash-in-advance (CIA) constraint. We
show that the long-term relationship between the income velocity of money
and the nominal growth rate of money supply depends not only on the formof the CIA constraint but also on the central bank’s stance of interest-rate
control rule.
Keywords and Phrases: velocity, an interest-rate control, endogenous
growth, cash-in-advance constraint
JEL Classification Numbers: O42, E52∗†

I am grateful to Kazuo Mino for his helpful comments on earlier versions of this paper.
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 563-

0043, JAPAN (email: ege010fs@mail2.econ.osaka-u.ac.jp)

1

Introduction

This paper examines the long-run effects of an endogenous monetary expansion via
the interest-rate control on income growth and on the velocity of money in thecontext of an AK growth model with a cash-in-advance (CIA) constraint.
In the existing literature, Suen and Yip (2005) and Chen and Guo (2008a) also
study the AK growth model with a CIA constraint. The main finding of those studies
can be summarized as follows. First, if the interetemporal elasticity of substitution
in consumption is less than one, the balanced growth path is unique and determinate,while there may exist dual balanced growth paths if the interetermporal elasticity
of substitution in consumption is higher than one. In the latter case, the balancegrowth path (BGP) with a higher growth rate is locally indeterminate and there is apositive relation between the growth rate of nominal money supply and the velocity
of money. Such a positive relationship is, however, empirically implausible. Chenand Guo (2008b) overcome this problem by assuming that the CIA constraint ismore effective on investment than on consumption. They justify this assumption
based on the recent increases in the consumer credit and in the cash holdings offirms.
These foregoing studies mentioned above assume that the central bank keeps
the growth rate of nominal money supply constant. However, many central banks
have shifted their policy stance from the base-money targeting to the interest-rate
control, we re-examine the long-run relation between monetary growth and velocity
of money under interest-rate control rules. Except for the monetary policy rule, we
employ the same analytical framework as Chen and Guo (2008b) use.
The present paper reveals that the relation between money growth and velocity
of money around the BGP depends on the stance of the monetary policy as well1

as on the form of CIA constraint. When the nominal interest rate responds to thecurrent rate of inflation alone, the velocity of money is negatively related to thegrowth rate of nominal money supply around the unique BGP, if the CIA constraint
is more binding for investment than consumption. However, if the interest rate
responds to the growth rate of income as well and it responds to inflation more than
one for one, then a lower velocity of money may be associated with a higher money
growth under the normal CIA constraint which is more effective for consumption
spending than for investment expenditure.

2

The Model

The representative household’s problem is to maximize a discounted stream of utilities

Z ∞0c1−σ −ρte dt,
1−σσ > 0,

ρ > 0,(1)subject toṁ = y − πm − c − ν + τ,k̇ = ν − δk,0 ≤ δ ≤ 1,

ψc c + ψν ν ≤ m,

0 < ψc ≤ 1,

k0 : given,0 ≤ ψν ≤ 1,

(2)
(3)
(4)

where c is consumption, ρ denotes the time discount rate, k is the household’s capitalstock, δ denotes the capital depreciation rate and σ is the inverse of the intertemporal elasticity of substitution in consumption. Moreover, ν is gross investment,Ṗ
y is output, π ≡is the rate of inflation, P the price level, and m denotes thePreal money balances that equal the nominal money supply M divided by P . The
household follows the budget constraint (2) and the dynamics of capital stock (3).

2The generalized CIA constraint (4) means that parts of consumption and gross investment must be financed by the household’s real money balances. The seigniorageis returned to households from the government as a lump-sum transfer so that the
government’s budget constraint is τ = ṁ + πm.
The production function is given by
y = Ak,A > 0,(5)and thus the market equilibrium condition for commodity, y = k̇ + δk + c, yieldsk̇= A − δ − z,
k

(6)

c
.
k
Following Taylor (1993), we assume that the monetary authority controls thewhere z ≡nominal interest rate by observing the real income as well as inflation. Since wedeal with a growing economy in which real income continuously expands, we assume
that the monetary authority changes the nominal interest rate in response not tothe level of income but to the growth rate of income. Although this formulation
is not exactly the same as in Taylor (1993), it is common that interest rate shouldbe raised in order to suppress the overheated economy. Specifically, we assume thefollowing control rule:µ
R = R(π, g) = π

∗

π
π∗

¶φ

µ

g
+ (A − δ) ∗g

¶η,φ ≥ 0,

φ 6= 1,

η ≥ 0,

A > δ, (7)where R is the nominal interest rate, and g denotes the growth rate of real income
given byg=ẏk̇
= = A − δ − z.
y
k

(8)

In addition, g ∗ > 0 represents the balanced-growth rate of income and π ∗ > 0
denotes the target rate of inflation. If φ > 1, the nominal interest rate rises more3

than one for one in response to a change in the rate of inflation. In this case, theinterest control rule is said to be active as to inflation. Conversely, the rule (7) withφ < 1 is defined as a passive monetary policy.
We focus on the economy’s BGP on which income, capital, consumption andreal money balances grow at a common rate. Combining (7) and (8) with the Fisher
equation,R − π = A − δ,(9)which implies that the real interest rate equals to the real rate of return to capital,
we obtainµA−δ+π =π

∗

π
π∗

¶φ

µ
¶η
A−δ−z
+ (A − δ)
.
g∗

(10)

Therefore, we see that the equilibrium rate of inflation depends on z, that is, π =π(z).
This type of monetary policy rule is also used in Fujisaki and Mino (2007), in
which money is introduced as the form of money-in-the-utility into an AK growthmodel. Fujisaki and Mino (2007) show that the response of the nominal rate ofinterest to the growth rate of income affects macroeconomic stability.

3

Dynamic System and Balanced Growth Path

We consider the case in which ψν is non-zero 1 . Denoting λm and λk as the shadowλk. As shown in Appendix,
prices of real money balances and capital, we define p ≡λm

1Since money is superneutral when ψν = 0, interesting result cannot be obtained even if ψc > 0.

Technically, p = 1 for all t in such a case so that the dynamic system is consisted by z alone andit does not depend on ψc .

4

we obtain the following dynamic system:
½ ·
¸
¾
1
ṗ A
ż =
l(p) + − ρ − δ − A + δ z + z 2 ,
σ
p
p
µ
¶
p2
1
ṗ =
+ δ − π(z) −p − A.
ψν
ψν

(11)
(12)

ψc − ψν. The dynamic equation of p (12) seems to be muchψc p − (ψc − ψν )simple than that in Chen and Guo (2008a, 2008b), but this is due to the specification
where l(p) = −of monetary policy. Therefore, the dynamic system in this paper is essentially thesame as in Chen and Guo (2008a, 2008b). Note that sign[l(p)] = −sign[ψc − ψν ],since p > 1 from the assumption that the CIA constraint is strictly binding inequilibrium.
In the BGP where ż = ṗ = 0 in (11) and (12), we obtain·¸
1 A− ρ − δ = A − δ − z ∗ > 0,
g =
σ p∗

(13)

p∗
1
A
+ δ − π∗ −− ∗ = 0.
ψν
ψν
p

(14)

∗

As above conditions show, the BGP is uniquely determined 2 , regardless of themagnitude of σ, which is in a marked contrast to the model with a fixed growth rateof nominal money supply.

2

From (12), we have two p∗ s, but one of them is negative, while another is positive and satisfiesp∗ > 1 since ṗ(p = 1; z = z ∗ ) = −(A − δ) − π ∗ < 0. Therefore, the number of plausible p∗ is only
one. Under the unique p∗ , we can give the nontrivial unique z ∗ . Determinacy of this unique BGP
is detailed in Appendix.

5

4

Velocity of Money

From now on, we investigate the relation between the velocity and the monetary
expansion rate. From (5) and ν = y − c, (4) is rewritten such thatm = (ψc − ψν )c + ψν Ak.

(15)

Differentiating this equation and defining µ as the nominal money growth rate, we
obtainṁ(ψc − ψν )ċ + ψν Ak̇
=µ−π =.
m(ψc − ψν )c + ψν Ak
Substituting

(16)

ż
ċ k̇
= − into (15), the endogenous nominal rate of monetary expanz
c ksion is:ż(17)µ = µ(z, p) = q(z) + A − δ − z + π(z),z(ψc − ψν )zwhere q(z) ≡is positive (resp. negative) if ψc > ψν (resp. ψc <(ψc − ψν )z + ψν A
ψν ). Under the interest-control rule, the nominal growth rate of money supply is
endogenously determined.
Since the growth rate of real money supply equals that of real income aroundṁ
the BGP, that is,= g ∗ , we derivemµ(z ∗ ) = A − δ − z ∗ + π(z ∗ ).

(18)

From (10),
η
A−δ
,
(19)
A − δ − z∗ φ − 1
·
¸
η
0 ∗
and therefore sign[π (z )]=signis satisfied around the BGP. As a result, we
φ−1
can obtain
η
A−δ
− 1,
(20)
µ0 (z ∗ ) = π 0 (z ∗ ) − 1 =A − δ − z∗ φ − 1
π 0 (z ∗ ) =6

Table 1: The Relation between Velocity and Money Supply
ψc > ψνψc < ψνψc = ψνη > 0, φ > 1, high A−δ
g∗

−

+

0

η > 0, φ > 1, low A−δ
g∗

+

−

0

η > 0, 0 < φ < 1

+

−

0

η=0

+

−

0

and thusµ0∗sign [µ (z )] = signηA − δ − z∗
−
φ−1
A−δ

¶
.

(21)

Note that g ∗ = A − δ − z ∗ . The income velocity of money around the BGP isrepresented byV ∗ (z ∗ ) =

y∗
A
=
,
∗
m
(ψc − ψν )z ∗ + ψν A

(22)

implying thatsign[V ∗ 0 (z ∗ )] = −sign[ψc − ψν ].

(23)

Combining (20) and (22), we can describe the effect of money supply on velocity
around the BGP as in Table 1. In this table, + means a positive relation, − meansa negative relation and 0 indicates that there is no relation.

5

Discussion

We consider intuitive implication of the results.
From (19), two effects of economic growth on the nominal expansion rate ofmoney supply can be seen. First, a decrease z ∗ yields a higher balanced-growthrate g ∗ . The second is the change of the inflation rate via the interest-control rule.
When the growth rate of income increases, the central bank should raise the nominal
7

interest rate to stabilize economy. However, since the net real rate of return to capitalis constant due to the assumption of AK technology, the real interest rate also should
be kept constant by adjusting the rate of inflation to satisfy the Fisher equation 3 .
This is achieved by depressing inflation if the monetary policy is generalized andactive, because the decline in nominal interest rate is larger than that in the inflation
rate. Therefore, the expansion rate of nominal money supply may fall. Otherwise,
the rise of economic growth does not generate the fall of the inflation rate so thatthe nominal growth rate of money supply increases.
We consider the result in (22). In light of AK technology, decreasing z ∗ generν∗ates the same magnitude of positive movement in the investment-capital ratio ∗ .
kTherefore, if ψc > ψν , velocity of money which means the ratio of capital to realmoney balances becomes larger. Conversely, when ψc < ψν , the negative relation
between the velocity and the growth rate of income is produced.
Taylor (1993) suggests that the observable policy stance of the Federal Reserve
may be described by setting φ = 1.5 and η = 0.5. Moreover, it is plausible toconsider that real investment expenditures for machines, factories and housing areless constrained by cash holdings than consumption spending. Therefore, we focus
on the case under which η > 0, φ > 1, and ψc > ψν . When the ratio of the net real
rate of return on capital A − δ to the balanced-growth rate g ∗ = A − δ − z ∗ is higher
A−δ
φ−1
<
, the negative relation between the nominal
enough to satisfy 0 <ηg∗money expansion and the income velocity of money holds, regardless of whether
or not the economy displays sunspot-driven fluctuations around the BGP. Since
φ−1A−δA − δ > g ∗ , the condition 0 < 1 =<holds for φ = 1.5 and η = 0.5.
ηg∗This result is different from Chen and Guo (2008b), who show that the negative

3

When the interest rate is controlled by the rate of inflation alone, this channel is not effective.

8

relation emerges only on the determinate BGP if ψc > ψν . Consequently, we canconclude that both the form of the CIA constraint and the central bank’s policy
stance are important for the relation between and velocity and money expansion.

References[1] Chen, S-H., and Guo, J-T. (2008a) ”Velocity of Money, Equilibrium(In)determinacy and Endogenous Growth”. Journal of Macroeconomics 30,
1085-1096. .
[2] Chen, S-H., and Guo, J-T. (2008b) ”On the growth and velocity effects ofmoney”, Economics Bulletin 5, 13, 1-7.
[3] Fujisaki, S. and Mino, K. (2007) Generalized Taylor Rule and Determinacy ofGrowth Equilibrium. Economics Bulletin 5, 11, 1-7.
[4] Meng, Q., and Yip, C. K. (2004) Investment, interest rules, and equilibrium
determinacy. Economic Theory 23, 868-878.
[5] Suen, M-H. and Yip, C-K. (2005) ”Superneutrality, Indeterminacy and Endogenous Growth” Journal of Macroeconomics 27, 579-595.
[6] Taylor, J., B. (1993) ”Discretion versus policy rules in practice” CarnegieRochester Conference Series on Public Policy 39, 195-214.

9

Appendix
The first-order conditions of the household’s maximization problem arec−σ = λm + ψc ζ,(24)

λk − λm = ψν ζ,(25)

λ˙k = (ρ + δ)λk − Aλm ,(26)

λ˙m = (ρ + π)λm − ζ,(27)

ζ(m − ψc c − ψν ν) = 0,

ζ ≥ 0,m ≥ ψc c + ψν ν,(28)

together with the transversality conditions lim e−ρt λmt mt = 0 and lim e−ρt λkt kt =
t→∞

t→∞

0, where λm and λk are the shadow prices of real money balances and capital, respectively, and ζ represents the Lagrange multiplier for the CIA constraint (4). In thefollowing, we assume that the CIA constraint (4) is strictly binding in equilibrium,
and thus ζ > 0 for all t. Using (23) through (26), we obtain the following dynamic
equations:

·
¸
ċ
1
ṗ A
=
l(p) + − ρ − δ ,
c
σ
p
p
A
λ˙k
=ρ+δ− ,
λk
p
µ
¶
λ˙m
1
p
= ρ + π(z) +
− .
λm
ψν
ψν

From these equations and (6), the dynamics equations (11) and (12) are derived.
We linearize the dynamic system (11) and (12) around the BGP to obtain:
 
 
  
ẑ 
ż  żz żp  ẑ (29)  = J  ,
 =
p̂
p̂
ṗz ṗpṗ
where ẑ ≡ z − z ∗ and p̂ ≡ p − p∗ . The elements of matrix J (28) are the following:
¯
· ∗
¸
µ 0 ∗
¶
∂ ż ¯¯l(p )ṗz
π (z )l(p∗ )
∗
żz =
=
+1 z = −+ 1 z∗,∂z ¯BGP
σp∗
σ
10

¯
∂ ż ¯¯z∗z ∗ p∗ (ψc − ψν ) + Aψc ψν
∗
∗
żp =
=
[l(p
)
ṗ
p
−
A]
=
−
,
p
∂p ¯BGP σ(p∗ )2σp∗ ψν [ψc p∗ − (ψc − ψν )]¯∂ ṗ ¯¯ṗz == −π 0 (z ∗ )p∗ ,∂z ¯BGP¯∂ ṗ ¯¯
A
p∗
=
+
> 0.
ṗp =
∂p ¯BGP p∗ ψνThe trace and determinant of J are respectively given by:µ¶π 0 (z ∗ )l(p∗ )A
p∗trJ = żz + ṗp = −+ 1 z∗ + ∗ + ,
σ
p
ψν
·
¸
z ∗ Aψν + (p∗ )2 Aπ 0 (z ∗ )detJ = żz ṗp − żp ṗz = ∗
.
−
p
ψν
σ

(30)
(31)

Since z and p are jump variables, if trJ > 0 and detJ > 0, then the BGP is totally
unstable so that the economy always stays on the BGP, that is, the equilibrium
path is determinate. Otherwise, the equilibrium path is indeterminate. That is,
the economy always stays on the BGP, or endogenous income fluctuations driven
by sunspots are generated. Inspecting (29) and (30), we can find the following
proposition, which shows that the generalization of the Taylor rule and the mostgeneralized CIA constraint play a significant role in macroeconomic stability.
Proposition 1 In the case of 0 < ψc , ψν ≤ 1, equilibrium determinacy holds either
if (i) monetary policy rule responds only to the rate of inflation. or if (ii) ψc ≤ ψνand monetary policy is passive. Otherwise, BGP could be locally indeterminate.

11