Discussion Papers In Economics
And BusinessTechnology, Preference Structure, and the Growth Effect ofMoney Supply
Jun-ichi Itaya and Kazuo Mino
Discussion Paper 05-35

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTechnology, Preference Structure, and the Growth Effect ofMoney Supply
Jun-ichi Itaya and Kazuo Mino
Discussion Paper 05-35

December 2005

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Technology, Preference Structure, and the Growth Eﬀect ofMoney Supply∗
Jun-ichi Itaya†and Kazuo Mino‡
December 9, 2005

Abstract
This paper studies the growth eﬀect of money supply in the presence of increasing
returns and endogenous labor supply. By using a simple model of endogenous growthwith a cash-in-advance constraint, it is shown that the growth eﬀect of money supply
depends on the specifications of preference structures as well as on the production
technology. Either if the production technology exhibits strong non-convexity or if
the utility function has a high elasticity of intertemporal substitution, then there may
exist dual balanced-growth equilibria and the impact of a change in money growthdepends on which steady state is realized in the long run. It is also shown that there
is no systematic relationship between the growth eﬀect of money supply and local
determinacy of the balanced growth path.
JEL Classification Numbers: E31, E52, O.42.
Keywords: monetary growth, indeterminacy, increasing returns, non-separable
utility.

∗

We are grateful to an anonymous referee for the comments that have been very helpful for revising thepaper. We also thank Chong Kee Yip and Shin-ichi Fukuda for their useful comments on earlier versionsof this paper. This reseach is financially supported by Grant-in-Aid for Scientific Research (# 1653017).
†
Graduate School of Economics and Business Administration, Hokkaido University, Kita 9, Nishi 7,
Sapporo, 060-0809 Japan, (phone: 81-11-706-2858, e-mail: itaya@econ.hokudai.ac.jp)‡
Corresponding Author, Graduate School of Economies, Osaka University, 1-7 Machikaneyama,
Toyonaka, 560-0043 Japan (phone: 81-6-6850-5232, e-mail:mino@econ.osaka-u.ac.jp)

1

Introduction

The growth eﬀect of money supply is one of the central research concerns in the money andgrowth literature. In particular, if the economy allows endogenous growth and if money
is not supernuetral, then monetary policy may have profound eﬀects on the long-term
behavior of the economy. Most of the existing studies on money and endogenous growth
have concluded that in economies with infinitely-lived agents, a higher money growthdepresses the long-run growth rate of the economy. The literature has demonstrated
that this conclusion holds in a wide variety of monetary endogenous growth models: see,for example, Howitt (1990), De Gregorio (1993), Jones and Manuelli (1995), Marquisand Leﬃtt (1991 and 1995), Pecorino (1995) and Mino (1997). The mechanism thatgenerates the negative relationship between monetary and real growth in the infinitelylived, representative agent economy has an intuitive appeal: a higher growth of money
supply raises the inflation tax on the rate of returns to physical (and human) capital, sothat investment will be reduced to yield a lower growth rate of the economy.1
The purpose of this paper is to re-examine the fundamental observation mentioned
above. We add two extensions to the conventional models employed by the existing studies
on money and endogenous growth. First, the model economy examined in this paper issubject to increasing returns to scale. Second, we use a more general utility function inwhich consumption and leisure can be non-separable. In the real business cycle studies,it has been known that both production technology and preference structure are crucial
for policy impacts as well as for the dynamic behavior of the economy. We demonstratethat this conclusion can be confirmed in the monetary growth models as well. Namely,
both the production technology and the preference structure may play essential roles indetermining the growth eﬀect of money supply. The analytical framework of our discussion
is based on the model examined by Benhabib and Farmer (1994) that has been frequently
used to discuss sunspot-driven business cycles. We introduce money into an endogenous
growth version of the base model via a cash-in-advance constraint. Since the model allows
1

In overlapping generations models with endogenous growth, however, inflation and growth may bepositively correlated: see, for example, Mino and Shibata (2000). Itaya and Mino (2003) and Jha et al.
(2002) examine endogenous monetary growth models based on transactions cost approach to money.

1

labor-leisure choice, money is not superneutral in the long-run equilibrium, even though
the cash-in-advance constraint applies to consumption alone.2
Our main finding is that the growth eﬀect of money supply hinges on the degreeof increasing returns characterizing production technology as well as on the elasticity ofintertemporal substitution in consumption. More specifically, if the elasticity of intertermporal substitution in consumption is lower than one and if production externalities are
relatively small, we obtain the conventional results, that is, the balanced-growth path isuniquely given and a rise in money growth reduces the long-term growth rate. In contrast, if the elasticity of intertemporal substitutability is smaller than one but the degreeof increasing returns is high enough, then there may exist dual balanced-growth paths.
In this case an increase in money growth decreases (increases) the growth rate on thebalanced-growth path with a higher (lower) growth rate. Those results are, however, reversed if the elasticity of intertemporal substitution in felicity exceeds one. Under such apreference characterization, if the degree of increasing returns is relatively small, then dualbalanced-growth paths may exit. If this is the case, a rise in monetary growth increases(decreases) the long-term growth rate in the high-(low-) growth equilibrium. Moreover, if
both the degree of increasing returns and the elasticity of intertemporal substitutability
are high, the balanced-growth path can be unique and a higher money growth accelerates
long-run growth.
In addition to the growth eﬀect of money supply, we investigate dynamic propertiesof the model economy out of the balanced-growth equilibrium. We find that, as well as
in the real business cycle models without money, the uniqueness of transitional path ofthe monetary economy heavily depends upon specifications of production technology andpreference structure. For example, in the presence of unique balanced-growth equilibrium,
the converging equilibrium path is uniquely determined when both the degree of increasing returns and the elasticity of intertemporal substitutability are relatively small. By
2

It is to be noted that Fukuda (1996) also introduces money into the Benhabib and Famer model witha separable utility function. Since we use a more general preference structure, our discussion involves
Fukuda’s main conclusions as special cases. Since Fukuda (1996) mainly focuses on the case of exogenous
growth in which long-term growth cannot be sustained, our study overlaps a part of his paper that treats
the case of endogenous growth.

2

contrast, when either the degree of increasing returns or the elasticity of intertemporal
substitution in consumption is high, a continuum of converging paths emerges around thebalanced-growth equilibrium. Additionally, it is shown that if there are dual balancedgrowth equilibria, one of them is determinate and the other is indeterminate. Whichbalanced-growth path satisfies determinacy again depends upon the specifications of technology and preferences. Based on those findings, we consider the relationship between thedeterminacy of equilibrium and the growth eﬀect of money supply.
The rest of the paper is organized as follows. Section 2 sets up an analytical framework
for the subsequent discussions. Sections 3 contains our main discussions. It considers thesteady-state eﬀects of a change in the money growth rate as well as the local dynamic
behavior of the balanced-growth under alternative conditions on the parameter values
characterizing the technology and preferences. Section 4 presents intuitive implications ofour main findings. Section 5 concludes the paper.

2

The Analytical Framework

2.1

Production

In formulating the production side of the economy, we follow Benhabib and Farmer (1994).
There are many identical firms whose number is normalized to one. The production
technology of each firm is specified as
y = ka l1−a k̄ α−a ¯lβ−1+a , 0 < a < 1, α > a, β > 1 − a,where y is output, k is capital and l is labor. Here, k̄ α−a and ¯lβ−1+a represent positive
external eﬀects associated with the capital and labor of the economy at large. We assume
that the private technology satisfies constant returns, but the social technology involving
the external eﬀects exhibits increasing returns to scale.
The final goods and factor markets are competitive, so that the rate of return tocapital, r, and the real wage rate, w, are equal to the marginal products of private capitaland labor, respectively:r = aka−1 l1−a k̄ α−a ¯lβ−1+a ,w = (1 − a) k a lk̄1−a ¯lβ−1+a .
3

Since we have assumed that the total number of firms is one, it always holds that k̄ = k and¯l = l in equilibrium. In this paper we focus on the situation in which endogenous growthis sustained by capital externalities. This means that α = 1, and hence the aggregate
production function at the social level is
y = klβ(1)

and the equilibrium levels of return to capital and real wage rate are respectively expressedas2.2

r = alβ ,(2)w = (1 − a) klβ−1 .

(3)Households

There is no population growth and the number of households is also normalized to be one.
The representative household maximizes a discounted sum of utilities
Z ∞e−ρt u (c, l) dt, ρ > 0U=0subject toȧ = ra + wl − c + τ − im,a = k + m,
c ≤ m.
In the above, c is consumption, a total asset, m real money balances, i the nominalinterest rate and τ denotes lump-sum transfers from the government. We assume thatthe government does not issue debt so that the total wealth of the household consists ofcapital and money holdings. In this paper, we assume that the cash-in-advance constraint
is applied to consumption spending alone.
The instantaneous utility function, u (c, l) , is assumed to be monotonically increasingin consumption and decreasing in labor. In what follows we use a specific form of function
such thatu (c, l) ≡c1−σ
Λ (l) ,
1−σ

σ > 0, σ 6= 1,(4)where Λ (l) is strictly positive and it satisfies the following conditions:
Λ0 (l)< 0,1−σ

Λ00 (l)< 0,1−σσΛ (l) Λ00 (l) + Λ0 (l)2 < 0.
1−σ
4

(5)

These conditions ensure that the felicity function decreases with labor and it is strictly
concave in consumption and labor.
To obtain the necessary conditions for an optimum, let us define the current-value
Hamiltonian function:H=c1−σΛ (l) + λ [ra + wl − c + τ − im] + μ (m − c) + γ (a − k − m) ,1−σwhere λ is the costate variable of a, and μ and γ are Lagrange multipliers. If we focus on
an interior solution, the necessary conditions for an optimum are given by the following:∂H/∂c = c−σ Λ (l) − (λ + μ) = 0,(6)c1−σ 0Λ (l) + λw = 0,
1−σ

(7)

∂H/∂l =

∂H/∂m = −λi + μ − γ = 0,(8)

∂H/∂k = −γ = 0,(9)λ̇ = ρλ − ∂H/∂a = λ (ρ − r) − γ,(10)

together with the transversality condition: lim t+∞ aλe−ρt = 0 and the initial condition
on a.

2.3

Money Growth and Market Equilibrium

Since this paper does not focus on comparison of the alternative money supply rules, we
assume the simplest money supply regime. Namely, the growth rate of nominal stock ofmoney is kept constant and the newly created money is distributed to the households as
lump-sum transfers. Denoting the growth rate of money supply by θ, we see that the realmoney balances behave in accordance withṁ = m (θ − π) .

(11)

The government’s flow budget constraint is thus given by τ = θm.
The market equilibrium condition for the final good is
y = c + k̇.

5

(12)

For notational simplicity, we assume that capital does not depreciate. Finally, the nominalinterest rate is determined by the Fisher equation:i = r + π,

(13)where π denotes the rate of inflation.

2.4

The Dynamic System

Equations (8) and (9) yield μ/λ = i. Thus using (6), (7), (8) and (9) , we obtain−w
c Λ0 (l)=.
1 − σ Λ (l)
1+i

(14)

This states that the marginal rate of substitution between labor and consumption is equal
to the relative price between leisure and consumption. Notice that when the cash-inadvance constraint is binding, an additional unit of consumption needs an additional
unit of real money balances so that the eﬀective price of consumption is one plus thenominal interest rate, i, which represents an opportunity cost of money holding. Therefore,w/ (1 + i) expresses the real wage rate in terms of the eﬀective price of consumption. Using(2) , (3) , (13) and (14) , we find that the rate of inflation is expressed as
π = − (1 − a) (1 − σ)

Λ (l) lβ−1x − alβ − 1 ≡ π (x, l) ,
Λ0 (l)(15)

where x ≡ k/c.
As we focus on an interior equilibrium, it holds that c = m for all t ≥ 0. Hence, from(11) and (15) consumption changes according toċ= θ − π (x, l) .
c

(16)

Equations (3) and (7) give the following:−c1−σ 0Λ (l) = λ (1 − a) klβ−1. .
1−σLogarithmic diﬀerentiation of both sides of the above equation with respect to time yields(1 − σ)l˙ λ̇ k̇l̇ċ+ η (l) = + + (β − 1) ,
c
l
λ k
l

6

where η (l) ≡ Λ00 (l) l/Λ0 (l) > 0. Therefore, using (9), (10), (12) and (16), we obtain
∙
¸
1
1
l˙
β
=
(σ − 1) (θ − π (x, l)) + (1 − a) l + ρ −,(17)lη (l) + 1 − βxin which we assume that η (l) + 1 − β 6= 0. Additionally, by use of (1) , (12), (16) andẋ/x = k̇/k − ċ/c, we can deriveẋ1= lβ − − θ + π (x, l) .
x
x

(18)

To sum up, (17) and (18) constitute a complete dynamic system with respect to l andx (≡ k/c) .

3

Growth Eﬀect of Money Supply

3.1

The Balanced-Growth Equilibrium

From (17) and (18) the steady-state levels of l and x satisfy the following conditions:(σ − 1) (θ − π (x, l)) + (1 − a) lβ + ρ −
lβ −

1
= 0,
x

1
− θ + π (x, l) = 0.
x

(19)
(20)

The steady-state conditions given above show that on the lanced-growth path capital consumption and real money balances grow at a common rate, while the level of employment
stays constant over time. Manipulating (15) , (19) and (20), we obtain
Λ (l) lβ−1
Λ0 (l)
∙ µ
¶
¸
´
i
h³
1 β ρ
a β ρ
l +
a 1−
l + +θ+1 .
=
1−
σ
σ
σ
σ
− (1 − a) (1 − σ)(21)

If this equation has a positive solution, then there exists a steady-state value of l. In order
to conduct a graphical analysis of (21) , let us define the following functions:Λ (l) lβ−1,
Λ0 (l)¶¸∙ µh³a´ β ρi
1 β ρG (l) ≡ 1 −l +l + +θ+1 .
a 1−
σ
σ
σ
σ
F (l) ≡ − (1 − a) (1 − σ)A positive solution of F (l) − G (l) = 0 gives the steady-state level of l.
7

(22)

(23)

Since the capital share of income, a, may be about 0.3 in reality, if σ < a, we should
assume that the intertemporal elasticity of consumption, 1/σ, takes an implausibly highvalue. Thus, in order to focus on economically plausible cases, we assume that σ > a.
Inspecting the graph of G (l) function defined above, we can easily confirm that if σ >1 (resp. a < σ < 1) and if F (l) is monotonically decreasing (resp. increasing) for all
feasible l, then F (l) − G (l) = 0 has a unique positive solution. Otherwise, there may exist
two solutions at most. The exact shape of F (l) cannot be determined without further
specifications. Therefore, we use the following function as a typical example to obtainclear results:
Λ (l) =

1
(1 − l)χ(1−σ) , 0 < χ < 1,
χ(24)which is defined on l ∈ [0, 1] . Since 1−l expresses leisure, (24) means that the instantaneous
felicity of the household is given by a standard Cobb-Douglas function of consumption andleisure in such a way thatu (c, l) =χ−1 (c (1 − l)χ )1−σ
.
1−σ

(25)

Note that this function fulfills the concavity conditions in (5) for σ > χ/ (1 + χ) . Since wehave assumed that σ > a, in what follows we assume that σ > σ̂ ≡ max (χ/ (1 + χ) /χ, a).
Given the above specification, we find that F (l) is written as
F (l) =

1
(1 − a) (1 − l) lβ−1 .
χFigures 1 (a) and (b) depict the graphs of F (l) and G (l) under σ > 1. Figure 1 (a) shows
the case of β < 1. In this situation, F (l) is monotonically decreasing and satisfies thatliml→+0 F (L) = +∞ and F (1) = 0. In addition, G (l) is a monotonically increasing andstrictly concave function for l ≥ 0. Thus F (l) − G (l) = 0 has a unique, positive solution.
Figure 1 (b) assumes that σ > 1 and β > 1. Under these conditions, F (0) = F (1) = 0hi
and sign [F 0 (l)] = sign 1 − β1 − l . In addition, G (l) is increasing, strictly convex and
G (0) = (ρ/σ) (ρ/σ + θ + 1) > 0. Thus if F (l) − G (l) = 0 has a solution, it involves tworoots for l ≥ 0.
Figures 2 (a), (b), (c) and (d) display the graphs of G (l) and F (l) for σ̂ < σ < 1. In
i1/β
h
¡¢
.
these cases, the graph of G (l) is inverse-U shaped and G ¯l = 0, where ¯l = ρ+σ(θ+1)
a(1−σ)

Thus when β < 1, there may exist dual steady-state levels of l if ¯l < 1, while the steady8state is uniquely determined if ¯l > 1: see Figures 2 (a) and (b). On the other hand, in thecase of β > 1, F (l) − G (l) = 0 has a unique positive solution if ¯l < 1, and there are twosolutions at most if ¯l > 1 : see Figures 2 (c) and (d).

3.2

Eﬀects of Monetary Expansion

From (1) , (12) , (19) and (20) the balanced-growth rate and the steady-state rate of
inflation are respectively given by
g=

´
1³ β
al − ρ ,σ

π =θ−g =θ+1(ρ − alβ ),
σ

(26)
(27)

where g denotes the balanced growth rate of y, c, k and m. Hence, the balanced-growthrate increases with l and the rate of inflation decreases with l. This implies that thebalanced-growth path with a higher employment level attains a higher growth rate and alower rate of inflation, while the other steady state with a lower employment exhibits alower growth rate and a higher rate of inflation.
Considering the steady-state characterization displayed above, we can easily examine
the eﬀects of an increase in the rate of money growth, θ, on the steady-state rates ofincome growth and inflation. Notice that an increase in θ causes an upward shift of thegraph of G (l) . Since the graph of F (l) is unaﬀected by a change in θ, it is easy to establish
the following comparative statics results:
Proposition 1 If functions F (l) and G (l) given by (22) and (23) satisfies G0 (l) > F 0 (l)at the steady state, a rise in money growth rate lowers the balanced-growth rate. If G0 (l) <F 0 (l) at the steady state, an increase in money growth raises the balanced-growth rate.
In the context of graphical analysis of G (l) and F (l) functions, we find that if G0 (l) >F 0 (l) , the negative relation between θ and l holds in the balanced-growth equilibrium. In
contrast, if G0 (l) < F 0 (l) is satisfied, the steady-state level of l (so the balanced-growthrate, g) is positively related to the money growth rate, θ. It is worth emphasizing that thesegraphical relations do not depend on our specification of Λ (l) function. Since the shapes ofG (l) and F (l) functions are characterized by the parameters concerning preferences and9technology, the above propositions clearly demonstrates that the growth eﬀect of money
supply relies critically on the preference structure as well as on the production technology.
If Λ (l) function is given by (24) , then in the presence of a unique balanced-growthpath, a rise in money growth rate, θ, lowers the long-term growth rate of income andraises the rate of inflation if β < 1 : see Figures 1 (a) and 2 (b). If β > 1, then, as
shown by Figure 2 (b), a higher θ increases the growth rate of income. If σ > 1 and if
dual balanced-growth paths exist, then a rise in θ depresses the growth rate of income
and increases the rate of inflation at the high-growth equilibrium, while it increases thegrowth rate of income at the low-growth equilibrium: see Figure 1 (b). If σ̂ < σ < 1, theopposite results hold (i.e. Figure 2 (b)). Table 1 summarizes these findings. In this table,
High and Low respectively denote the balanced growth path (BGP) with a higher and
i1/β
h
.3
lower growth rate. In addition, σ̂ = max (χ/ (1 + χ) , a) and ¯l = ρ+σ(θ+1)
a(1−σ)σ>1σ̂ < σ < 1 and ¯l < 1σ̂ < σ < 1 and ¯l > 1

dual BGPs
β<1unique BGP,

unique BGPHigh : dg/dθ > 0

dg/dθ < 0dg/dθ < 0
Low : dg/dθ < 0dual BGPs
β>1dual BGPs
unique BGPHigh : dg/dθ < 0High : dg/dθ < 0dg/dθ > 0

Low : dg/dθ > 0

Low : dg/dθ > 0
Table 1

3.3

Dynamics

Next, let us examine the dynamic property of our model economy around the balancedgrowth equilibrium. When we linealize the dynamic system consisting of (17) and (18) atthe stationary point, the coeﬃcient matrix of the resulting linearized system is given by
∙
¸ ⎤
⎡
¤
£
l
1
l
β−1
(1 − σ) π x (l, x) + 2 ⎥(1 − σ) π l (l, x) + (1 − a) βl⎢
η (l) + 1 − βx
J = ⎣ η (l) + 1 − β £
⎦,
¤
1
β−1
x βl
+ π l (l, x)xπ x (l, x) +
x
(28)
3

The definition of ¯l states that l̄ is relatively large when θ, ρ and σ have large values and it is relatively

small when β and a are high.

10where l and x indicate their steady-state values. Since we treat an endogenous growthmodel with an Ak technology, neither of the initial values of l nor x (≡ k/c) are predetermined. This means that if the linearized system is completely unstable around thesteady state (i.e. the stationary point is a source), the economy can always stay on thebalanced growth path so that local determinacy of equilibrium is established. However, if
the steady state of the dynamic system is either a saddle-point or a sink, then there exists
a continuum of converging paths and hence local indeterminacy emerges. Keeping thesefacts in mind, we can show the following analytical results:
Proposition 2 The balanced-growth equilibrium has a local saddlepoint property, either
if η (l) + 1 − β > 0 and F 0 (l) > G0 (l) or if η (l) + 1 − β < 0 and F 0 (l) < G0 (l) at thesteady state.
The proof of this proposition is shown in Appendix 1 of the paper. It is worth emphasizing that the stability property is related to the sign of η (l) + 1 − β, while the characterization of the steady state and the growth eﬀect of money supply depend critically on thesign of 1−β. Therefore, if we consider both the stability and steady state characterization,
we should classify the cases into β < 1, 1 < β < 1 + η (l) and β > 1 + η (l) . Applying theseresults to the case of Cobb-Douglas utility given by (25) , we find that local determinacyof equilibrium at each steady state may be summarized in Table 2. In the table D and I
respectively indicate local determinacy and indeterminacy of BGPs. Appendix 2 presents
the detail to drive the results shown in the table.
σ>1σ̂ < σ < 1 and ¯l < 1σ̂ < σ < 1 and ¯l > 1

dual BGPs
β<1unique BGP,

unique BGPHigh : I

D?
Low : ?

dual BGPs
1 < β < 1 + η (l)dual BGPs
unique BGPHigh : ?

High : ?
I

Low : I

Low : I

dual BGPs
β > 1 + η (l)dual BGPs
unique BGPHigh : I

High : I
D

Low : D

Low : D11

Table 2

Tables 1 and 2 demonstrate that there are various patterns of the relationship between
the growth eﬀect of money supply and determinacy of equilibrium. That is, we may havethe counter-intuitive, positive relation between money growth and real growth, even if
determinacy is satisfied around the balanced-growth path. At the same time, the standard
negative relation between money growth and real growth can be established in the longrun equilibrium with local indeterminacy. For example, when σ > 1 and β < 1, the uniquebalanced growth path exhibits local determinacy. In contrast, although there is a uniquebalanced-growth equilibrium, it is locally indeterminate if 1 < β < 1 + η (l) and σ < 1.
However, if β exceeds 1 + η (l) , the long-run equilibrium becomes locally determinate. In
these cases, the growth eﬀect of money supply is negative if σ > 1 and β < 1, while it ispositive if β > 1 and σ < 1.

3.4

Separable Utility

In order to highlight the role of the non-separable utility in our discussion, it is useful to
examine a simple model with an additively separable utility function. Suppose that the
instantaneous utility is given byu (c, l) = log c − Φ (l) ,(29)

where Φ0 (l) > 0 and Φ00 (l) > 0. It is easy to confirm that the dynamic equation of l inthis case may be obtained by setting σ = 1 in (17) , while the behavior of x is the same as(18). Thus the complete dynamical system in the case of separable utility consists of thefollowing diﬀerential equations:
l̇
l
ẋ
x

∙
¸
1
1
β
=
(1 − a) l + ρ −,η (l) + 1 − βx1= lβ − − θ + π (x, l) ,x

where η (l) = Φ00 (l) l/Φ0 (l) (> 0) and the rate of inflation is given by
π (l, x) =(1 − a) lβ−1x − alβ − 1.
Φ0 (l)

12

As a result, the steady-state conditions under which l̇ = ẋ = 0 can be reduced to thefollowing equation whose solution gives the steady state value of l:hi(1 − a) lβ−1
β
=
(1
+
ρ
+
θ)
(1
−
a)
l
+
ρ
.
Φ0 (l)(30)

We see that the left-hand side of (30) is monotonically decreasing (resp. increasing) , if
η (l) + 1 − β > 0 (resp. η (l) + 1 − β < 0) for all l > 0. Figures 3 (a) and (b), in which£¤F̂ (l) ≡ (1 − a) lβ−1 /Φ0 (l) and Ĝ (l) ≡ (1 + ρ + θ) (1 − a) lβ + ρ , show the existence ofthe steady-state level of l. As Figure 3 (a) depicts, if η (l) +1 − β > 0, the balanced-growthequilibrium is uniquely given. If η (l) +1 − β < 0, then there are at most two steady states:
see Figure 3 (b). A typical example of Φ (l) function isΦ (l) =

lη+1
,
η+1

where η is a positive constant. This specification of disutility of labor has been frequently
employed in the real business cycle literature. Since η us constant for all l > 0 in thiscase, the global argument given above always holds.
It is also to be noted that in the case of unique balanced-growth path, it is locallydeterminate. In the case of dual steady states, the BGP with a lower growth rate (a lower
level of l) is locally indeterminate and that with a higher growth rate (a higher level ofl) is locally determinate. The positive relation between the balanced-growth rate and themonetary expansion rate, θ, holds only at the low-growth BGP in the case of dual steadystates. These results mean that, as compared to the case of non-separable utility, in themodel with a separable utility function the degree of labor externalities, i.e. the magnitudeof β, plays a predominant role in determining the growth eﬀect of money supply as wellas the determinacy of equilibrium.
To sum up, we have shown:
Proposition 3 If the utility function is given by (29) and if β < 1 + η (l) for all l > 0,then the balanced-growth equilibrium is unique and globally determinate. If β > 1 + η (l)for all l, there may exist dual steady states at most. In the presence of dual steady states,
the balanced-growth equilibrium with a lower growth rate is locally determinate and theone with a higher growth rate is locally indeterminate, provided that dual balanced-growthequilibria exist.
13

4Discussion

4.1

The Roles of Technology and Preference

In this section we present intuitive implications of the roles of production technology andpreference structure in determining the growth eﬀect of money supply. To do so, we
assume that the economy initially stays on the balanced-growth path and that there is apermanent rise in the growth rate of nominal money supply, θ.
Notice that the steady-state rate of nominal interest rate satisfies i = r+π = (σ − 1) g+
ρ+θ, because r = σg +ρ and π = θ −g in the balanced-growth equilibrium. First, consider
the case of separable utility given by (29) . As was shown, in this case the growth eﬀect
of money depends mainly on the magnitude of β. Noting that under the separable utility
assumption i = ρ + θ on the balanced-growth path, we can rewrite (30) ashi (1 − a) lβ−1.
Φ0 (l) (1 − a) lβ + ρ =
1+ρ+θ

(31)

The left-hand side of (31) is the steady-state expression for the marginal rate of substitution between consumption and labor, while the right-hand side shows the steady-state levelof real wage rate in terms of eﬀective price.4 We may interpret the left- and right-handsides of (31) as the steady-state representations of the labor supply and labor demand
curves, respectively. Figure 4 illustrates the labor supply and demand curves in (l, w)
space. A permanent rise in θ increases the steady-state rate of nominal interest rate,
which causes a downward shift of the labor demand curve. If β < 1, the labor demand
curve has a negative slope. Thus an increase in θ reduces the steady-state value of employment, l, which lowers the balanced growth rate because of the relation given by (26):see panel (a) in Figure 4. In words, a higher rate of inflation caused by an increase in themoney growth rate raises the opportunity cost of money holding, and hence the eﬀective
cost of consumption also rises due to a tighter cash-in-advance constraint. Consequently,
the households substitute consumption with leisure, which reduces labor supply so thatthe balanced-growth rate declines.
If β > 1, then the labor demand curve has a positive slope. As shown by Proposition
4

More precisely, the left hand side of (31) is the ratio of the marginal rate of substitution and capitalstock, while the right hand side expresses the ratio of the (eﬀective) real wage and capital stock.

14

1, in this case there are two steady states and it is straightforward to confirm that thelong-run labor demand curve is steeper (resp. flatter) than the labor supply curve at
the steady state with a lower (resp. higher) level of l.5 Therefore, an downward shift ofthe labor demand curve caused by a rise in θ increases (resp. decreases) the steady-statelevel of l at the low-(resp. high-) growth steady state. Figure 3 (b) shows the case oflower-growth steady state where a rise in the nominal interest rate caused by a higher θenhances employment and thus income growth. By contrast, Figure 3 (c) depicts the caseof high-growth steady state where a higher θ lowers employment and the balanced-growthrate.
We should note that the growth eﬀect of money supply in the case of separable utility
essentially has the same eﬀect of labor income taxation in the absence of cash-in-advanceconstraint. If there is no money and if real factor income is subject to distortionary
taxation, the budget constraint for the household’s optimization problem is replaced withk̇ = (1 − τ k ) rk + (1 − τ l ) wl − c,where τ k and τ l denote the tax rates on capital and labor income, respectively. Given thisconstraint, the optimization condition (14) becomescΦ0 (l) = (1 − τ l ) w.
Thus the labor market equilibrium condition corresponding to (31) is modified as[(1 − a) lβ + ρ]Φ0 (l) = (1 − τ l ) (1 − a) lβ−1 .
Replacing 1 − τ l with (1 + θ + ρ)−1 in the above equation, we see that a rise in therate of labor income tax, τ l , yields the same growth eﬀect as that of an increase in money
growth rate, θ. Hence, when the preference structure satisfies additive separability between
5£¤
The slope of the graph of LHS in (31) is (1 − a) βlβ−1 Φ0 (l) + (1 − a) lβ + ρ Φ00 (l) , and that of RHS

is (1 − a) (β − 1) lβ / (1 + ρ + θ) . Remember that the graphs given in Figure 3 satisfies
F̂ 0 (l)≡(1 − a) (1 − β) lβ−2 /Φ0 (l) − (1 − a) lβ−1 Φ00 (l) /φ02 (l)Ĝ0 (l)=(1 r + θ) (1 − a) βlβ−1 .

Thus, using Ĝ (l) = F̂ (l) , we find that if F̂ 0 (l) > Ĝ0 (l) at the intersection, the slope of demand curve is
steeper than that of the labor supply curve, and vise versa.

15

consumption and labor, the inflation tax plays the role exactly analogous to that playedby the labor income taxation.6
When the utility function is not separable between consumption and labor, the nominalinterest rate in the steady state is i = (σ − 1) g + ρ + θ. By use of g = (1/σ) (r − ρ) , wefind that the relation between the long-term nominal interest rate and the steady-statelevel of employment is¶µ
1
alβ + ρ + θ.
i= 1−
σ

(32)

This equation demonstrates that if σ ≥ 1, the nominal interest rate increases with theemployment level (so the balanced-growth rate) in the steady state, while it decreases
with l if σ < 1. In view of (32) , the equality between the marginal rate of substitution
and the real wage rate in the steady state can be written as−(1 − a) lβ−11
Λ0 (l) [(1 − a) lβ + ρ] =
,
1−σ
1+i

(33)

where i is given by (32) . Since the case of σ ≥ 1 yields essentially the same outcomes as
those obtained in the model with separable utility, we examine the case of σ̂ < σ < 1.
In order to employ graphical interpretation displayed in Figure 4, we take i as a given
parameter. Inspecting the graph of (33) reveals that if β > 1, the labor demand curve
has a positive slope. Therefore, if the labor supply curve has a positive slope and if thedemand curve is steeper than the supply curve, the situation is depicted by Figure 4 (b).
Again, a rise in the money growth rate, θ, shifts the labor demand curve downward, which
increases the steady-state level of employment and long-term growth rate.
We should note that if σ̂ < σ < 1, the labor supply curve may have a negative slope.
As shown by Figures 4 (d) and (e), if the labor demand curve is less steeper than the labor
supply curve, then a downward shift of the labor demand due to a rise in θ increases thesteady state levels of employment and growth rate. This gives an intuitive implication
for the positive growth eﬀect of money supply obtained in the case of σ̂ < σ < 1.7 Since
67Here, we implicitly assume that income tax is not levied on leisure.
The eﬀect of a change in θ on the steady-state rate of nominal interest is
µ
¶
1
di
dl
=1+ 1−
βlβ−1 .
dθσdθThus if dl/dθ < 0, the nominal interest rate in the steady state increases when σ < 1. If dl/dθ > 0, we
could obtain di/dθ < 0. Figures 4 (d) and (f) ignore this possibility.

16

under the assumption of non-separable utility the steady-state level of nominal interest
rate is endogenized and thus the eﬀect of inflation tax is not so straightforward as that in
the case of separable utility under which the long-term nominal interest rate, i = ρ + θ,
is independent of the level of l. These observations imply that the preference structure
plays an essential role in determining the growth eﬀect of money supply when the utility
function is not additively separable between consumption and labor.

4.2

Equilibrium Determinacy and the Growth Eﬀect of Money

We have shown that if there are dual balanced-growth paths, at least one of them is locally indeterminate in the sense that there are a continuum of converging paths aroundthe balanced-growth equilibrium.8 Moreover, even when the balanced-growth path isuniquely given, there may exist a continuum of converging paths around the balancedgrowth equilibrium. The presence of real indeterminacy means that sunspot-driven expectations aﬀect the dynamic behavior of real variables. In this case, even if the money
growth rate is kept constant, a change in the anticipated rate of inflation produced bya sunspot-driven change in expectations may have real eﬀects. Such a disturbance, ofcourse, will not aﬀect the balanced-growth equilibrium, but it may bring the economy
out of the balanced-growth path. Since there exists a continuum of converging paths atleast around the balanced-growth equilibrium, the economy may converge to the original
position. During the transition towards the balanced-growth path, the growth rate of
the economy may change. Therefore, in the presence of equilibrium indeterminacy, any
extrinsic uncertainty about the money growth rate may have the growth eﬀect.
As emphasized before, an important conclusion of our discussion is that there is no
systematic link between the growth eﬀect of money supply and the local determinacy ofequilibrium. According to the corresponding principle in the sense of Samuelson (1983,
8

Many authors have demonstrated that complex preference structures may yield indeterminacy of equi-

librium in the monetary dynamic models based on the money-in-the-utility function formulation: see, forexample, De Fiore (2000), Carlstrom and Fuerst (2001), Matheny (1998) and Matsuyama (1991). Since
we assume the presence of a cash-in-advance constraint on consumption alone, the relation between consumption and real money balances is extremely simple in our setting. Hence, indeterminacy caused by thepreference structure in our model mainly stems from the labor supply behavior rather than money demand
behavior of the households.

17

Chapter 9), one may conjecture that the counter-intuitive policy eﬀect, i.e. a rise in money
growth rate promotes economic growth despite the presence of cash in advance constraint,
is associated with indeterminacy of equilibrium. We have, however, seen that a higher
growth of money supply may have a positive eﬀect on the long-term growth rate of income
even when the balanced-growth equilibrium satisfies local determinacy. Conversely, thenegative relation between monetary expansion and real growth would prevail even in thepresence of equilibrium indeterminacy.
It should be also noted that if the balanced-growth equilibrium exhibits local indeterminacy, we may not observe a one-to-one relation between inflation and growth in thelong run. If expectations-driven fluctuations may emerge, the economy easily diverges
from the balanced-growth path without any monetary disturbance. Therefore, when theequilibrium indeterminacy holds, the long-run relationship between money supply andreal growth derived by the comparative statics exercise does not help so much to identify
the actual relationship between inflation and growth. Although empirical researches on
the relation between inflation and growth have not yet reached a consensus, both in the
cross-country regressions and in the time-series analyses the following finding has been
widely accepted: among the group of low inflation countries, it is hard to find statisticallymeaningful relationships between the growth rate of income and the rate of inflation.9
One of the interpretations of these findings is that there is no long-term relation between
money and growth at least in low-inflation countries, that is, money is superneutal in
the long-run. Alternatively, we may also reason that such an absence of clear relation
between inflation growth can be attributed to equilibrium indeterminacy emerged in the
low-inflation economies. Since our model with nonseparable utility can yield various combinations of the growth eﬀect of money supply and equilibrium determinacy, it may provide
us with a useful analytical framework to interpret various empirical results within a single
model.
9See, for example, Barro (1996) and Bruno and Easterly (1996). Temple (2000) surveyed the empirical aswell as theoretical studies on inflation and growth. He summarized controversies and unresolved problems
in this field.

18

5Conclusion

In the context of a simple model of endogenous growth with a cash-in-advance constraint,
we have explored the growth eﬀect of money supply in the long-run equilibrium. We haveshown that the magnitude of the elasticity of intertemporal substitution in consumption
as well as the degree of increasing returns are crucial for determining the growth eﬀect ofmoney supply. We have also confirmed that local indeterminacy of equilibrium hinges onthese two key parameters. At the same time, we have found that there is no systematic
link between the growth eﬀect of money supply and the equilibrium determinacy aroundthe balanced-growth path.
The present paper has assumed that the growth rate of nominal money supply is kept
constant. The recent studies on indeterminacy in monetary dynamic models emphasize
the role of money supply rule. In particular, the interest rate control rules may generateindeterminacy even in models with convex production technologies and separable utility
functions.10 The existing investigations on this issue, however, have been mostly concernedwith models without capital accumulation. Clarifying the relation between technology,
preference and money supply rules in the presence of capital formation would deserve
further investigation.11
Appendices
Appendix 1 Proof of Proposition 2
From (15) and (22) the rate of inflation is written as
π (l, x) = − (1 − a) (1 − σ)

Λ (l) lβ−1x − alβ − 1 = F (l) x − alβ − 1,
Λ0 (l)

implying that π l (l, x) = F 0 (l) x − aβlβ−1 and π x (l, x) = F (l) . Hence, the determinant of
the coeﬃcient matrix J can be written as
½
µ
¶
i¾
xl
1
σ h
β−1
β−1
0
(σ − a) βl+ F (l) x .
det J =
F + 2 − 2 (1 − a) βl
η (l) + 1 − βxx((A1))
10
11

See, for example, Benhabib et al. (2000).
Some authors have considered the relationship between the interest rate rules and determinacy ofequilibrium in models with capital formation: see Carlstrom and Fuerst (2005), Meng and Yip (2004) and
Itaya Mino (2004).

19

Note that
µ
¶
i
1
σ h
(σ − a) βl
F + 2 − 2 (1 − a) βlβ−1 + F 0 (l) x
x
x
½∙µ
¶
µ
¶¸
¾
1
1
1
σ
β−1
0
1−
F (l) x +1−βl
− F (l) .
((A2))
=
x
σ
x
σ
h³
¤
¢
a´ β ρi£ ¡Since G (l) is given by G (l) ≡ 1 −l +a 1 − σ1 lβ + σρ + θ + 1 , we obtain:
σ
σ
¶
¶
∙³
µ
µ
a´
1 β ρ
0
β−1
1−
G (l) = βll + +θ+1
a 1−
σ
σ
σ
µ
¶ ³³
¸
´
´
1
a β ρ
+a 1 −.
((A3))
1−
l +
σ
σ
σ
¡
¢
By using the fact that F (l) x = π + alβ + 1, π = θ − g = θ − (1/σ) alβ − ρ and
β−1

1/x = lβ − θ + π = (1 − a/σ) lβ + ρ/σ, in equation (A2) we can show
¶
µ
¶
µ
1
1
1
F (l) x +
1−
1−
σ
x
σ
∙ µ
¶
¸
µ
¶
³
´
a
1 β ρ
1 ³³
a´ β ρ´
= 1−
a 1−
l +
.((A4))
l + +θ+1 +a 1−
1−
σ
σ
σ
σ
σ
σ

From (A1) , (A2), (A3) and (A4), we finally obtain:
det J =
Therefore, it holds that

¤
£ 0
σl
G (l) − F 0 (l) .
η (l) + 1 − β£¤sign det J = sign [η (l) + 1 − β] G0 (l) − F 0 (l) ,which confirms the proposition.

Appendix 2 Local stability when the utility function is given by (25) .
First, notice that when the utility function is (25) , we obtain
η (l) =Λ00 (l) ll= [1 − χ (1 − σ)].
Λ0 (l)1−l

β−1Hence, if β > 1, then β > 1 + η (l) for 0 ≤ l ≤ β−χ(1−σ). In the following, when considering
β−1
. The
the case of β > 1 + η (l) , we restrict our attention to the region of 0 ≤ l ≤ β−χ(1−σ)

trace of the coeﬃcient matrix J is written as
Tr J =h³´i1l(1 − σ) xF 0 (l) − aβlβ−1 + (1 − a) βlβ−1 + xF (l) +η (l) + 1 − βx20

Thus in the case of Figure 1 (a) (σ > 1 and β < 1) in which F 0 (l) < 0, we see that on
the balanced-growth path det J > 0 and Tr J > 0. This means that the steady state isa source and local determinacy is established. If β > 1 + η and σ > 1 (Figure 1 (b)),
the high-growth steady state where G0 > F 0 , we obtain: det J < 0 and thus the steadystate is a saddlepoint, implying that the BGP is locally indeterminate. In contrast, inthe low-growth steady state it is easy to confirm that Tr J > 0 and det J > 0, so that
local determinacy holds. In the case of Figure 2 (a), we cannot determine the sign of Tr
J without imposing further restrictions. Since det J > 0 in this case, the steady state canbe either a source or a sink. Similarly, in Figure 2 (b) the stability property of the lowgrowth steady state cannot be determined unless we use numerical examples, while thehigh growth steady state is a saddlepoint (i.e. locally indeterminate) because det J < 0.
Note that if 1 < β < 1 + η (l) holds at the steady state, the sign of Tr J cannot be
determined at the high-growth steady state both for the cases σ > 1 and σ < 1. Since
det J > 0 in these steady states, they may be either sinks or sources. By contrast, thelow-growth steady state satisfies det J < 0 and thus it is locally indeterminate.

21

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22

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23

G(l)

F(l)
l



(a) σ > 1, β <1

G(l)

F(l)



(b) σ >1, β > 1

Figure 1

l

F(l)

F(l)

G(l)

G(l)

l

l



(b) σ < 1, β <1, l < 1, l >1

(a)

l



l

σ < 1, β < 1

F(l)
G(l)

F(l)
G(l)



l

l

l
F  σ < 1, β > 1, l < 1

E σ < 1, β > 1, l > 1

Figure 2l (l )G

 (l )Fl
C β < 1 + η

 (l )G

 (l )
Fl(b) β > 1 + η(KIWTG

w

w

labor demandlabor supply

labor demandlabor supply

l

(a)

(e)

w

l

w
labor demandlabor supply

labor demandlabor supply

(b)

l

(f)

w
labor supply(c)labor demandl(KIWTG

l