ISSN (Print) 0473-453X

Discussion Paper No. 1076

ISSN (Online) 2435-0982

EFFECTS OF MONETARY POLICYIN A MODELWITH CASH-IN-ADVANCE CONSTRAINTS ON
R&D AND CAPITAL ACCUMULATION

Daiki Maeda
Yuki Saito

Revised June 2020.
February 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Eﬀects of Monetary Policy in a Model with
Cash-in-Advance Constraints on R&D and Capital
Accumulation∗
Daiki Maeda†

Yuki Saito‡

June 17, 2020

Abstract
To examine the eﬀect of monetary policy on economic growth, we formulate an
endogenous growth model with cash-in-advance constraints on R&D and capital
accumulation as endogenous growth engines. Within this framework, we show thatthe relationship between economic growth and the nominal interest rate can be
an inverted-U shape. Moreover, we demonstrate that the welfare-maximizing levelof the nominal interest rate is larger than the growth-rate-maximizing level of thenominal interest rate.

Keywords: Capital accumulation; CIA constraint; Endogenous growth; Monetary policy;
R&D
JEL classiﬁcation: E52; O31; O40∗

We would like to thank Hiroki Fukai, Koichi Futagami, Tatsuro Iwaisako, Takaaki Morimoto,
Yoshiyasu Ono, Akihiko Yanase, and workshop participants at Chukyo University, Kobe University,
and Osaka University for their helpful comments and suggestions on a previous version of this paper. Allremaining errors are ours. This research is ﬁnancially supported from the Japan Society for the Promotion of Science through Grants-in-Aid for Scientiﬁc Research (S) Grant No. 15H05728 and International
Joint Research Promotion Program (Osaka University).
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka, 5670047, Japan. E-mail addresses: u035866j@alumni.osaka-u.ac.jp‡
Corresponding author: School of Economics, Chukyo University, 101-2 Yagoto-honmachi, Showa-ku,
Nagoya, Aichi 466-8666, Japan; E-mail: yuki.saito.r@gmail.com; yuki.saito@mecl.chukyo-u.ac.jp.

11Introduction

The relationship between economic growth and monetary policy has been one of the more
important issues among economists for several years. In response, a number of theoreticalstudies have investigated the eﬀect of monetary policy on economic growth and haveshowed that monetary policy in an economy with a cash-in-advance (CIA) constraint on
investment aﬀects the long-run growth rate. Two strands of literature are evident in thesestudies. The ﬁrst strand employs a model in which households need to accumulate money
before they consume and purchase capital; thus, in this model, there are CIA constraints
on both consumption and capital accumulation (i.e., investment). For example, early
work by Stockman (1981) and Abel (1985) demonstrates that the steady-state capitalstock inversely relates to the rate of inﬂation.
The second strand develops an R&D-based growth model in which R&D ﬁrms needto borrow money before they pay their costs and households need to accumulate money
before they consume. Accordingly, in this model, there are the CIA constraints on bothconsumption and R&D activity (i.e., investment). In particular, Chu and Cozzi (2014)
show that when overinvestment in R&D takes place, the welfare-maximizing level of thenominal interest rate is positive.1 According to these two strands, investment activities,
especially capital accumulation and R&D activity, face a CIA constraint, and thus capital accumulation and R&D activity, which are important driving forces of endogenous
growth, are aﬀected by monetary policy through the CIA constraint. However, existingstudies employ either the CIA constraint on capital accumulation or that on R&D activity.
Therefore, the purpose of the present paper is to formulate an endogenous growth modelwith CIA constraints on both capital accumulation and R&D activity and to examine theeﬀects of monetary policy on economic growth and welfare.
To formulate an endogenous growth model in which there are CIA constraints on
1

Chu and Cozzi (2014) is the ﬁrst paper to formulate an R&D-based growth model with CIA constraints on consumption and R&D activity. Subsequent to Chu and Cozzi (2014), a number of theoreticalstudies employ an R&D-based growth model with a CIA constraint on R&D activity. See, for example,
recent work by Chen (2018), Chu, Cozzi, Furukawa, and Liao (2019) and Hori (2020).

2

both capital accumulation and R&D, we extend Iwaisako and Futagami’s (2013) model inwhich there are two endogenous growth engines, one for R&D and another for capital accumulation, so that capital accumulation and R&D activity both face a CIA constraint.2
The setting of the CIA constraint on investments draws on Chu and Cozzi (2014) (i.e., weassume that R&D ﬁrms and capital-producing ﬁrms need to borrow money from households to ﬁnance part of their costs in advance). In our model, the nominal interest rate
is a monetary policy instrument, and we show that there are cases where the relationshipbetween economic growth and the nominal interest rate is an inverted-U shape (i.e., thenominal interest rate which maximizes the growth rate is positive).
In addition, in our model we assume that the strength of the CIA constraint on the
R&D sector and that on the producing capital sector diﬀers. Therefore, an increase in
the nominal interest rate stimulates investments in that sector where there is less need toborrow money. If this sector is then more productive, the growth rate increases when thenominal interest rate increases. As a result, the growth-maximizing nominal interest rate
is positive; thus, we observe an inverted-U-shaped relationship between economic growth
and the nominal interest rate. Moreover, conducting welfare analysis, we demonstratethat the welfare-maximizing level of the nominal interest rate is larger than the growthrate-maximizing level of the nominal interest rate.
Several studies have already obtained an inverted-U-shaped relationship between economic growth and the nominal interest rate in an R&D-based growth model with a CIA
constraint on R&D.3 Chu, Cozzi, Furukawa, and Liao (2017) and Chu, Cozzi, Fan, Furukawa, and Liao (2019) develop a Schumpeterian growth model with endogenous entry ofheterogeneous ﬁrms. In their model, only the R&D sector faces a CIA constraint. In contrast, the present paper develops an endogenous growth model with two growth engines,
2

In Iwaisako and Futagami’s (2013) model, the correlation between the growth rates of the number ofintermediate goods and physical capital is negative owing to the speciﬁcation of the technologies for R&D
and capital production. In their model, both R&D and capital production use labor. This contrasts with
Romer (1990) and Rivera-Batiz and Romer (1991), which have two growth engines, namely, a varietyexpansion type of R&D and capital accumulation.
3
For instance, Arawatari, Hori, and Mino (2017) suggest that the relationship between inﬂation andgrowth becomes nonlinear in an R&D-based model with the heterogeneous R&D abilities of agents.

3

R&D and capital, and both the R&D and capital-producing sectors face diﬀerent strengthsof the CIA constraint. Therefore, by deriving an inverted-U-shaped relationship between
economic growth and the nominal interest rate through the diﬀerent mechanisms, we provide a useful complement to these existing analyses. Huang, Yang, and Zheng (2018) and
Zheng, Huang, and Yang (2019) develop an endogenous growth model with two R&D
sectors. In their model, R&D ﬁrms in both sectors face a CIA constraint. In contrast, thepresent paper derives an inverted-U-shaped relationship between economic growth andthe nominal interest rate in a diﬀerent setting and thus provides a useful complement to
Huang et al. (2018) and Zheng et al. (2019).
The present paper also relates to the literature on monetary policy and economic
growth in an endogenous growth model with R&D and (physical or human) capital accumulation. As an example, Chu, Lai, and Liao (2019) develop a monetary hybrid endogenous growth model in which R&D and capital accumulation are both engines ofendogenous growth. However, they consider a CIA constraint only on consumption. Elsewhere, Chu, Ning, and Zhu (2019) explore the growth and welfare eﬀects of monetary
policy in a scale-invariant Schumpeterian growth model with endogenous human capital accumulation, and consider CIA constraints on both consumption and R&D investment. Likewise, Gil and Iglésias (2020) employ an endogenous growth model with R&D
and physical capital accumulation to examine the eﬀect of monetary policy on economic
growth.4 Consequently, that work considers the CIA constraints on R&D activity andthe manufacturing of intermediate goods. However, none of these studies consider CIA
constraints on both capital accumulation and R&D activity.5
The remainder of the paper is organized as follows: Section 2 provides our model andSection 3 addresses the market equilibrium. Section 4 explains the equilibrium path and
4

Gil and Iglésias (2020) extend Rivera-Batiz and Romer’s (1991) model such that the growth ratesof the number of intermediate goods and physical capital are determined proportionally. In contrast, thepresent paper extends Iwaisako and Futagami’s (2013) model and thus the growth rates of the numberof intermediate goods and physical capital exhibit a negative correlation. See also footnote 2.
5
In Gil and Iglésias (2020), the CIA constraint on the manufacturing of intermediate goods aﬀects
capital accumulation because capital is a production factor of these intermediate goods and thus capital
accumulation and R&D activity are aﬀected by the CIA constraints. However, they do not provide an
inverted-U-shaped relationship between economic growth and monetary policy.

4Section 5 details the eﬀect of the nominal interest rate. Section 6 concludes.

2

The model

To examine the relationship between economic growth and monetary policy, we develop
an endogenous growth model with CIA constraints. In this model, there are two enginesof endogenous growth, namely, capital accumulation and R&D, with investments in bothconducted using labor. We assume that R&D ﬁrms and capital-producing ﬁrms need toborrow money from households to ﬁnance part of their costs in advance (i.e., there are
CIA constraints on both R&D and capital accumulation). A continuum of intermediate
goods then produces the ﬁnal goods, with the varieties of intermediate goods expanding
because of R&D activity. Labor and capital produce each intermediate good.

2.1

Households

In the economy, there are a ﬁxed number of identical households. We normalize thetotal number of households to unity. We deﬁne L as the size of the population of eachhousehold. Each member of the household has one unit of labor and supplies her or hislabor inelastically at each point of time. Then, the sum of labor supply is L at each point
in time. The representative household maximizes lifetime utility over an inﬁnite horizon
given by:∫ ∞e−ρt log C(t)dt,(1)0where C(t) denotes the consumption of ﬁnal goods and ρ is the subjective discount rate.
The representative household maximizes (1) subject to the following budget constraint:ȧ(t) + ṁ(t) = r(t)a(t) + i(t)b(t) + w(t)L + τ (t) − C(t) − π(t)m(t),(2)

where a(t) is the real value of assets excluding money, m(t) is the real money balance heldby households, b(t) is the amount of real money borrowed from each member of households

5

by entrepreneurs (who engage in R&D and producing capital) to ﬁnance investment, τ (t)is a lump-sum transfer from the government, r(t) is the rate of return on assets or thereal interest rate, i(t) is the return on lending money, w(t) is the real wage rate, and π(t)is the inﬂation rate. The CIA constraint is given by:b(t) ≤ m(t).

(3)

As there is no CIA constraint on consumption, the following familiar Euler equation
characterizes the solution to the preceding dynamic maximization problem:Ċ(t)= r(t) − ρ.
C(t)

(4)

The dynamic optimization requires the following relationship:6i(t) = r(t) + π(t).

(5)

This equation is the Fischer equation. Therefore, i(t) is the nominal interest rate.

2.2

Firms producing ﬁnal goods

The production function of ﬁnal goods is represented by a Dixit and Stiglitz (1977) typefunction:
[∫
N (t)

] α1
x(j, t)α dj

Y (t) =

, 0 < α < 1,

(6)0

where Y (t) is the output of ﬁnal goods, N (t) is the number of intermediate goods available,x(j, t) is the input of intermediate good j ∈ [0, N (t)], and 1/(1 − α) > 1 represents theelasticity of substitution between the intermediate inputs. Given we deﬁne the ﬁnal good6

See Appendix A for the detailed calculation.

6

as a numeraire, the solution of the ﬁnal good ﬁrm’s cost-minimizing problem is given by:x(j, t) = Y (t)p(j, t)− 1−α ,1(7)where p(j, t) is the price of intermediate good j. Substituting (7) into (6) yields:
[∫

] α1

N (t)

α
− 1−α

p(j, t)

1=

dj

.

(8)

0

2.3

Firms producing intermediate goods

The production of each intermediate good employs labor and physical capital. The production function of intermediate good j is speciﬁed as a Cobb–Douglas form:

x(j, t) = Al(j, t)γ k(j, t)1−γ ,(9)where x(j, t) is the output of intermediate good j, and l(j, t) and k(j, t) are the amountsof labor and capital used for the production of intermediate good j, respectively. Solving
the cost-minimizing problem yields the following unit cost function:c(w(t), q(t)) = A−1 γ −γ (1 − γ)γ−1 w(t)γ q(t)1−γ ,(10)where q(t) is the rental rate of capital. From Shephard’s lemma, the demand functions
for labor and capital can be derived as follows:∂c(w(t), q(t))γc(w(t), q(t))
x(j, t) =x(j, t)∂w(t)w(t)(1 − γ)c(w(t), q(t))∂c(w(t), q(t))
x(j, t) =x(j, t).
k(j, t) =
∂q(t)
q(t)
l(j, t) =

(11)
(12)

As in the growth model, each intermediate good ﬁrm can monopolistically supply with aproduct variety expansion.7 Therefore, the proﬁt-maximizing price of intermediate good7

See the canonical model in Romer (1990) and Grossman and Helpman (1991, Ch. 3).

7

j is given by:8p(j, t) =1c(w(t), q(t)).
α(13)

This equation implies that all intermediate goods ﬁrms charge the same price level (i.e.,p(j, t) = p(t)). Substituting (13) into (8) yields:
1−αc(w(t), q(t)) = αN (t) α .

(14)

Using (7), (13) and (14), we obtain the output and proﬁt of the intermediate good as
follows:[

1]− 1−α1
1x(t) =c(w(t), q(t))
Y (t) = Y (t)N (t)− α
α
(
)
1
Y (t)
Π(t) =− 1 c(w(t), q(t))x(t) = (1 − α)
,
α
N (t)

(15)
(16)

where we omit the goods index j because the intermediate ﬁrms are symmetrical from
(13).

2.4

R&D ﬁrms

In this subsection, we characterize the R&D ﬁrm’s behavior, and examine how the valueof the patent, v(t), is determined. v(t) is equal to the sum of the discounted present value
( ∫s
)
∫∞
of the proﬁt ﬂow after period t: i.e., v(t) = t Π(s) exp − t r(ν)dν ds. Diﬀerentiating
this equation with respect to time, t, we obtain the following equation:v̇(t) = r(t)v(t) − Π(t).

8

(17)

This paper does not consider patent policy (i.e., patent protection is complete). A number of studies
analyze the eﬀect of patent protection in R&D-based growth models. See, for example, early work by
Judd (1985), Li (2001), Goh and Olivier (2002), Iwaisako and Futagami (2003), Kwan and Lai (2003),
and O’Donoghue and Zweimuller (2004).

8

Rewriting (17) yields the familiar no-arbitrage condition: r(t)v(t) = Π(t) + v̇(t). Thisequation means that the return from holding one patent, r(t)v(t)dt, must be equal to thesum of the monopolistic proﬁt, Π(t)dt, and the capital gain or loss, v̇(t)dt.
The production function for R&D is given by:Ṅ (t) = (aN )−1 N (t)lN (t),(18)

where lN (t) is the labor input used in R&D activity, and aN > 0 is constant and represents
the reciprocal of the productivity of labor. In the present paper, we assume that the ﬁrm
must borrow money to ﬁnance the cost of a proportion of ηN ∈ [0, 1] before they invest in
R&D as in Chu and Cozzi (2014).9 ηN represents the strength of the CIA constraint on
R&D. Given the ﬁrm’s cost is w(t)lN (t), the amount of borrowing is ηN w(t)lN (t). Hence,
the total cost of the ﬁrm is w(t)lN (t) + i(t)ηN w(t)lN (t) = (1 + ηN i(t))w(t)lN (t). Using
this equation, the proﬁt of the ﬁrm is v(t)Ṅ (t) − (1 + ηN i(t))(aN /N (t))w(t)Ṅ (t). Because
we assume free entry, the following free-entry (or zero-proﬁt) condition must be satisﬁed:

v(t) = (1 + ηN i(t))

2.5

aN
w(t).
N (t)

(19)

Firms producing capital goods

In this subsection, we characterize the behavior of the ﬁrm producing capital, and examine
how the value of one unit of capital stock, vK (t), is determined. vK (t) is equal to the sumof the discounted present value of the proﬁt ﬂow after period t: i.e.,
( ∫s
)
∫∞
vK (t) = t q(s) exp − t r(ν)dν ds. Diﬀerentiating this equation with respect to time,t, we obtain the following equation:v̇K (t) = r(t)vK (t) − q(t).

This is also the familiar no-arbitrage condition.
9

That is, there is a CIA constraint on R&D.

9

(20)

The production function for producing capital is given by:K̇(t) = (aK )−1 K(t)lK (t),(21)

where lK (t) is the labor input used in producing capital, and aK > 0 is constant anddenotes the reciprocal of the productivity of labor. We also assume that the capitalproducing ﬁrm must borrow money to ﬁnance some proportion of the cost (i.e., there is a
CIA constraint on producing capital). When ηK ∈ [0, 1] denotes the cost of the proportion
needed to be borrowed with money before the ﬁrm can produce,10 the proﬁt of the ﬁrm isvK (t)K̇(t) − (1 + ηK i(t))(aK /K(t))w(t)K̇(t). Given that we assume free entry, we impose
the following free-entry (or zero-proﬁt) condition:vK (t) = (1 + ηk i(t))

2.6

aK
w(t).
K(t)

(22)

Money Authority

The monetary authority supplies money and transfers all of its (i.e., the seigniorage revenue) to each household equally in each period. Deﬁning M (t) as the nominal money
supply and P (t) as the ﬁnal good’s price, the lump-sum transfer is given by:τ (t) = [Ṁ (t)/M (t)][M (t)/P (t)] = [Ṁ (t)/M (t)]m(t), where M (t)/P (t) = m(t) is the definition of the real money balance. We assume that monetary policy is implemented bycontrolling the nominal interest rate denoted by i (i.e., the nominal interest rate is themonetary policy instrument determined by the monetary authority, and thus i(t) = i overtime). Therefore, the growth rate of the nominal money supply, Ṁ (t)/M (t), is determined
endogenously.

10

Given the ﬁrm’s cost is w(t)lK (t), the amount of borrowing is ηK w(t)lK (t). Thus, the total cost is
w(t)lK (t) + ηK i(t)w(t)lK (t).

103Market equilibrium

The households inelastically supply L units of labor in each period. Firms producing
intermediate goods, ﬁrms engaging in R&D, and ﬁrms producing capital goods all demand labor. Therefore, the equilibrium condition for the labor market is given by:∫ N (t)l(j, t)dj + lN + lK = L. Substituting (14) and (15) into (11), we obtain the fol0
lowing labor demand function of a representative ﬁrm producing one intermediate good:l(t) = αγY (t)/[w(t)N (t)].

(23)

Using this equation, (18), (21) and the equilibrium condition for the labor market, we
obtain:αγK̇(t)Ṅ (t)
Y (t)
+ aK
+ aN
= L.
w(t)
K(t)
N (t)

The equilibrium condition for the capital market is:∫ N (t)0(24)k(j, t)dj = K(t) because capitalis only used for the production of intermediate goods. Substituting (14) and (15) into
(12), we obtain: k(t) = α(1 − γ)Y (t)/[q(t)N (t)]. Moreover, substituting this equation
into the equilibrium condition for the capital market yields:α(1 − γ)Y (t)= K(t).
q(t)(25)

The ﬁrms providing R&D and those producing capital borrow money. The R&D ﬁrm borrows ηN w(t)lN (t) and the capital-producing ﬁrm borrows ηK w(t)lK (t). As the households
lend b(t) unit of money, the market for borrowing is given by ηN w(t)lN (t)+ηK w(t)lK (t) =b(t). Combining this equation, (18) and (21), we obtain:

ηK w(t)aKK̇(t)Ṅ (t)+ ηN w(t)aN= b(t).
K(t)
N (t)

11

(26)

From Appendix A, the households use all of their money held for lending to entrepreneurs.
Therefore, in the equilibrium, the following equation is satisﬁed:

b(t) = m(t).

(27)

The ﬁnal goods market-clearing condition is given by:

Y (t) = C(t).

4

(28)

Equilibrium pathWe deﬁne V (t) ≡ [v(t)N (t)]/(1 + ηN i) and VK (t) ≡ [vK (t)K(t)]/(1 + ηK i). From thefree-entry conditions for R&D, (19), and producing capital, (22), the equilibrium wage is
given by:w(t) =VK (t)
V (t)
=
.
aN
aK

(29)

Using (25), we obtain the aggregate rental, Q(t) ≡ q(t)K(t), as follows:
Q(t) = α(1 − γ)Y (t).

(30)

We substitute (29) into (24) to obtain:gN (t) =

L
aK
aK 1
−
− αγ
gK (t),YaN
aN VK (t) aN

12

(31)

where gN (t) ≡ Ṅ (t)/N (t), gK (t) ≡ K̇(t)/K(t) and VKY (t) ≡ VK (t)/Y (t)11 . Substituting
the proﬁt ﬂow, (16), and (29) into the no-arbitrage condition, (17), yields:
v̇(t)
aK
1
=r(t) − (1 − α).
v(t)aN (1 + ηN i)VKY (t)(32)

Moreover, substituting the aggregate rental, (30), into the no-arbitrage condition, (20),
we obtain:v̇K (t)1.
=r(t) − α(1 − γ)vK (t)(1 + ηK i)VKY (t)(33)

The nominal interest rate is the constant policy parameter because we assume that it is
the policy instrument of the monetary authority. Using (30) to (33), and i is constant,
we obtain the growth rates of V (t) and VK (t) as follows:
v̇(t)
aK
aK 1
V̇ (t)
1L
aK= gN (t) += r(t) − (1 − α)− αγgK (t),
+
−
Y
Y
V (t)
v(t)
aN (1 + ηN i)VK (t) aN
aN VK (t) aN(34)
V̇K (t)v̇K (t)1+ gK (t).
= gK (t) += r(t) − α(1 − γ)
VK (t)vK (t)(1 + ηK i)VKY (t)(35)

From (29), V̇ (t)/V (t) = V̇K (t)/VK (t) holds for all t. Combining (34) and (35), we canderive the growth rate of capital:1gK (t) =aK + aN

{
L+

[
)]}
(
α
1−γ
(1 − α)/αaN − aK+γ.
1 + ηN i
VKY (t) 1 + ηK i

(36)

From the market-clearing condition of the ﬁnal good, it follows that Ċ(t)/C(t) = Ẏ (t)/Y (t).
Substituting this into the Euler equation (4) yields gY (t) ≡ Ẏ (t)/Y (t) = r(t) − ρ. we use

11

Although the growth rates of variety and capital may be negatively related, it is not surprising. In
our model, the labor force input determines these rates. Therefore, if the labor input is concentrated in
the R&D (capital) sector, it correspondingly decreases in the capital (R&D) sector. This is consistent
with the argument in Iwaisako and Futagami (2013). In fact, the above case may arise in the equilibrium.

13

this equation, (35) and (36) to get:(V̇KY (t) =Lρ+aK + aN)
VKY (t) −aK
aK + aN[1−α+α1 + ηN i(

1−γ+γ
1 + ηK i

)]
.

(37)

From this equation, we obtain the following lemma.
Lemma 1. The path initially jumping to a unique balanced growth path (BGP) is themarket equilibrium path.
Proof. See Appendix B.
This lemma implies that if policy changes unexpectedly, the market equilibrium pathhas no transition dynamics because it immediately jumps to the unique BGP.

5

Eﬀects of the nominal interest rate

In this section, we examine the eﬀects of the nominal interest rate on the BGP, which is
the market equilibrium path. We ﬁrst derive the equilibrium value of VKY (t). Combining
V̇KY (t) = 0 and (37) yields:

VKY = aK1−α+α1+ηN i(1−γγ + 1+ηKi

L + (aN + aK )ρ

)
,

(38)

where VKY is the equilibrium value of VKY (t). Hereafter, we deﬁne the variable omitted t
as the equilibrium values (or the values on the BGP).
To start, we analyze the eﬀects of the nominal interest rate on the rate of growth of
the number of intermediate goods and capital. Using (31), (36), and (38), we obtain the14

growth rate of the number of intermediate goods and capital on the BGP as follows12 :

gN =

1 1−α
L + (aN + aK )ρ
() − ρ,
aN 1 + ηN i 1−α + α γ + 1−γ(39)

L + (aN + aK )ρ
α 1−γ() − ρ.
aK 1 + ηK i 1−α + α γ + 1−γ(40)

1+ηN igK =1+ηK i1+ηN i

1+ηK i

From these equations, we obtain the following lemma.
Lemma 2. Suppose that the nominal interest rate, i ≥ 0, is suﬃciently small. An increasein the nominal interest rate then enhances the growth rate of the number of intermediate
goods if (ηN /ηK ) < 1 − γ. Moreover, an increase in the nominal interest rate enhances
the growth rate of capital if (ηN /ηK ) > 1 + [(αγ)/(1 − α)].
Proof. Diﬀerentiating (39) and (40) with respect to the nominal interest rate, i, we obtainηN −ηKγηN + (1 − γ) (1+η2
(1 − α)[L + (aN + aK )ρ]
dgN
K i)
=−
{
[
]}2 ,
di
aN
1+ηN i1 − α + α γ(1 + ηN i) + 1+η
(1
−
γ)
Ki

α(1 − γ)[L + (aN + aK )ρ]
dgK
=−
{
di
aK
1+ηK i1+ηN iηK −ηN
αγηK + (1 − α) (1+η2N i)

(1 − α) + α [γ(1 + ηK i) + 1 − γ]

}2 .

(41)

(42)

If each condition is satisﬁed, the above equations are positive when i = 0 (i.e., dgN /di|i=0 >0 if (ηN /ηK ) < 1 − γ and dgK /di|i=0 > 0 if (ηN /ηK ) > 1 + [(αγ)/(1 − α)]).
We interpret Lemma 2 as follows. An increase in the nominal interest rate aﬀects
the growth rates of the number of intermediate goods (capital) through the following twochannels. First, a higher nominal interest rate makes R&D (capital-producing) activity
more costly through the CIA constraint and thus stiﬂes investment in the R&D sector
12Equations (39) and (40) imply that the growth rates of the number of intermediate goods andcapital diﬀer. This is not consistent with Gil and Iglesias (2020) and Howitt and Aghion (1998) given thediﬀerence in the speciﬁcation of the investment technologies in the R&D and capital-producing sectors.
In their model, R&D and capital production are through inputting ﬁnal goods under a linear technology.
Therefore, the growth rates of the ﬁnal goods, the number of intermediate goods, and capital are identicalin the BGP. However, in our model, R&D and capital production are through inputting labor. As a result,in the BGP in our model, the number of labor inputs for each sector can diﬀer. Therefore, the rate of
growth of the number of intermediate goods and capital can be diﬀerent, as in Chu, Lai, and Liao (2019).
See also footnote 4.

15(the capital-producing sector). Hence, a higher nominal interest rate drives down thegrowth rates of the number of intermediate goods (capital). This negative eﬀect positively correlates with the strength of the CIA constraint on R&D (capital-producing)
activity. Second, a higher nominal interest rate also makes capital-producing (R&D) activity more costly through the CIA constraint and thus stiﬂes investment in the capitalproducing sector (the R&D sector). This brings about a reduction in labor demandin the capital-producing sector (the R&D sector). Labor is then reallocated from thecapital-producing (R&D) sector to both the intermediate goods production sector andthe R&D (capital-producing) sector. This is because R&D and capital-producing activity
and the production of intermediate goods all use labor. Hence, a higher nominal interest
rate drives up the growth rates of the number of intermediate goods (capital) throughlabor reallocation.13 This positive eﬀect positively correlates with the strength of the
CIA constraint on capital-producing (R&D) activity. If the CIA constraint in the R&D(capital-producing) sector is relatively weaker than that in the capital-producing (R&D)
sector, the second positive eﬀect of the nominal interest rate on the growth rates of thenumber of intermediate goods (capital) can dominate the ﬁrst negative eﬀect. Hence, anincrease in the nominal interest rate stimulates the growth rate of the number of intermediate goods (capital) when the nominal interest rate is suﬃciently low. Lemma 2 implies
that an increase in the nominal interest rate can bring about resource reallocation fromthe investment sector with a relatively stronger CIA constraint to the investment sector
with a relatively weaker CIA constraint.
Next, we analyze the eﬀect of the growth rate of output. Output can be represented
by the unit cost function of the intermediate ﬁrms, (10). Using (15), we obtain:[∫
Y (t) =
0

N (t) (] α1α)− 1−α1c(w(t), q(t))
Y (t)α djα1= [αAγ γ (1 − γ)1−γ w(t)−γ Q(t)γ−1 K(t)1−γ ] 1−α Y (t)N (t) α .
13

1

(43)

This positive eﬀect comes from the speciﬁcation of the technologies for R&D and capital production.
See footnote 11.

16Substituting (29) and (30) into (43), we obtain the value of output as follows14 :Y (t) = A (αγaK )γ (VKY )−γ K(t)1−γ N (t) α .
1−α

(44)

Given VKY does not change on the BGP, the growth rate of output, gY ≡ Ẏ (t)/Y (t), is
given by gY = (1 − γ)gK + [(1 − α)/α]gN from (44). Substituting the growth rates, (39)and (40), we obtain:[]α (1 − γ)2 (1 − α)2 1
1
L + (aN + 1)ρ
() + constant termgY =+aK 1 + ηK i
αaN 1 + ηN i 1−α + α γ + 1−γ
1+ηN i
1+ηK i[]()
2
1 (1 − γ)21−α11
L + (aN + 1)ρ
=++ constant term,(1−α)/αaK 1 + ηK i
αaN 1 + ηN i
+ γ + 1−γ
1+ηN i

1+ηK i(45)
where constant term is the term that does not depend on the nominal interest rate. From
this equation, we obtain the following proposition:
Proposition 1. If(1 − γ)2 1 − α 1−> (<)0
aK
α aN

(46)

and
ηN
αγ
> (<)1 +ηK1−α( 1−α )2 1

2
+ (1−γ)αaNaK(1−γ)2
1
− 1−α
aK
α aN

(47)

are satisﬁed, the nominal interest rate maximizing the growth rate of output is positive.

14

The reason that VKY is included in (44) is as follows. In our model, the intermediate goods areproduced by inputting not only capital but also labor. Therefore, the production of the intermediate
goods decreases if the input of labor is decreased by an increase in wage. A decrease in the production ofintermediate goods reduces the output of the ﬁnal goods. The term VKY = VK (t)/Y (t) = aK w(t)/Y (t)in (44) captures this channel.

17

Proof. Diﬀerentiating (45) with respect to the nominal interest rate, i, we obtain:
dgY
= {[
di
(1−α)/α

L + (aK + aN )ρ
]
}2
1−γ
+
γ
+
(1
+
η
i)(1
+
η
i)
K
N
1+ηN i
1+ηK i{()
1−α1−γ 1−α 1
×(1 − γ)−(ηN − ηK )
α
aK
α aN
[
]}
(
)2
η
1
1
−
α
K
− γ (1 − γ)2 (1 + ηN i)2 +.
ηN (1 + ηK i)2
aK
α
aN

(48)

If conditions, (46) and (47), are satisﬁed, (48) is positive when i = 0. Furthermore, it iseasy to prove d2 gY /di2 < 0. Therefore, the nominal interest rate maximizing the growthrate of output is positive.
Proposition 1 implies that there is an inverted-U-shaped relationship between economic
growth and monetary policy (i.e., the nominal interest rate). We interpret Proposition 1
as follows. From (48), the necessary conditions for the nominal interest rate maximizing
the growth rate of output becoming positive are [(1 − γ)/aK ] − [(1 − α)/α](1/aN ) > (<)0and ηN > (<)ηK . First, we consider the case where [(1 − γ)/aK ] − [(1 − α)/α](1/aN ) > 0
and ηN > ηK .15 This condition is satisﬁed when γ and aK are smaller. This means that
there is advantage to producing capital because the marginal productivity of capital in
the ﬁnal good sector is high and the capital-producing cost in this case is low. Therefore,
when [(1 − γ)/aK ] − [(1 − α)/α](1/aN ) > 0, increasing capital production makes thegrowth rate of output increase. From (42), if ηN > ηK , there is a possibility of anincrease in the nominal interest rate increasing the growth rate of capital. Therefore,
there exists a case in which increasing the nominal interest rate increases the growth rateof output. Similarly, we obtain the reason why the growth rate of output increases when[(1 − γ)/aK ] − [(1 − α)/α](1/aN ) < 0. When there is an advantage to R&D, increasing
R&D makes the growth rate of output increase. From (41), if ηN < ηK , there is also15ηN (ηK ) is the strength of the CIA constraint on R&D (capital accumulation). According to Chu andCozzi (2014), empirical studies (e.g., Brown and Petersen 2009; Brown, Martinsson, and Petersen 2012;
Brown and Petersen 2015) support the argument that R&D investment is even more severely aﬀected byliquidity requirements than physical capital accumulation. Hence, in practice, ηN > ηK is likely to hold.

18

the possibility that increasing the nominal interest rate increased the growth rate of thenumber of intermediate goods. Then, the growth rate of output increases.
In the last part of this section, we discuss the welfare eﬀects. Given C(t) = Y (t),lifetime utility is given by:

W ≡

∫ ∞

e−ρt log Y (t)dt.

(49)

0

As the BGP is the market equilibrium path, Y (t) = Y (0)egY t . Therefore, we rewrite (49):
( )2
]
1 [
gY − ργ log VKY + constant term,
W =
ρ

(50)

where “constant term” does not depend on i. Diﬀerentiating (38) with respect to i yields
dVKY /di < 0. Therefore, “−ργ log VKY ” is increasing in the nominal interest rate, i. Thisimplies that the welfare-maximizing level of the nominal interest rate is larger than thegrowth-rate-maximizing level of the nominal interest rate. The reason is as follows. A
higher nominal interest rate increases the costs of these two types of investment throughthe CIA constraints, and thus leads to a lower level of investment and lower labor inputs
into these investments. This makes it possible to reallocate labor from investment toproduction. As a result, a higher nominal interest rate increases the production level
and thus improves welfare. The existence of the eﬀect as we explained brings about theresult that the welfare-maximizing level of the nominal interest rate is larger than thegrowth-rate-maximizing level of the nominal interest rate.

6

Conclusion

By formulating a simple endogenous growth model with CIA constraints on both capital accumulation and R&D activity, we have shown that the relationship between economic growth and the nominal interest rate as the monetary policy instrument can be aninverted-U shape depending on the strength of the CIA constraints. In the realistic case19

where capital accumulation and R&D activity, which are important driving force of endogenous growth, face CIA constraints, the growth-rate-maximizing level of the nominalinterest rate can be positive. This implies that monetary authority policymakers should
consider the strength of the CIA constraints on all investment sectors to maximize thegrowth rate.

AppendixA

The optimization of the household

We denote the Hamiltonian as follows:

H = ln C(t) + λ(t)(r(t)a(t) + i(t)b(t) + w(t)L + τ (t) − C(t) − π(t)m(t) − Im (t)) + κ(t)Im (t)+ ϕ(t)(m(t) − b(t)),(A.1)

where λ(t) and κ(t) are the co-state variables of assets and money, respectively,ṁ(t) ≡ Im (t), and ϕ(t) is the Lagrangian multiplier of the CIA constraint. The optimality
conditions are given by:1− λ(t) = 0
C(t)

(A.2)

Im (t) : −λ(t) + κ(t) ⋛ 0(A.3)

b(t) : i(t)λ(t) − ϕ(t) ⋛ 0(A.4)

a(t) : λ̇(t) − ρλ(t) = −r(t)λ(t)(A.5)m(t) : κ̇(t) − ρκ(t) = π(t)λ(t) − ϕ(t)(A.6)

ϕ(t) : ϕ(t)(m(t) − b(t)) = 0 and ϕ(t) ≥ 0(A.7)

C(t) :

Transversality condition (TVC) lim e−ρt λ(t)a(t) = lim e−ρt κ(t)m(t) = 0
t→∞

t→∞

20

(A.8)

From (A.2) and (A.5), we obtain the Euler equation (4). From (A.3), if the households
hold both assets and money, we obtain:

λ(t) = κ(t).

(A.9)

λ̇(t)
κ̇(t)
=
.
κ(t)
λ(t)

(A.10)

From (A.9), we get:Using (A.5), (A.6), (A.9) and (A.10), we obtain:

ϕ(t) = (r(t) + π(t))λ(t)(A.11)

Substituting (A.4) into (A.11), and if b(t) > 0, [i(t) − (r(t) + π(t))]λ(t) = 0. If assets are
held, λ(t) ̸= 0. Therefore, we get (5).

B

Proof of Lemma 1

First, we show that the BGP is unique. The diﬀerential equation given by (37) has aunique but unstable nonzero steady state given the following inequalities hold:LaKi ≥ 0, ρ +> 0 andaK + aN
aK + aN[1−α+α1 + ηN i(

1−γ+γ
1 + ηK i

)]
> 0.

(B.1)

This means that there is a unique BGP.
Second, we show that the path initially jumping to the BGP is the market equilibrium
path. Given all assets are held by households and (29) must be satisﬁed in equilibrium,)(aN
VK (t).
a(t) = VK (t) + V (t) = 1 +
aK

21

(B.2)

Using this equation, C(t) = Y (t), and (A.2), we rewrite the transversality condition(TVC) as follows:
−ρt

lim e

t→∞

1
C(t)

(
(
)
)
aN
aN
−ρt Y
1+
VK (t) = lim e VK (t) 1 +
= 0.
t→∞
aK
aK

(B.3)

At ﬁrst, we consider the case in which VKY < VKY (t), where VKY is the steady-state valueof VYK (t). In this case, VKY (t) diverges to positive inﬁnity. From (37), the growth rate ofe−ρt VKY (t)[1 + (aN /aK )] is given by:

[
(
)]
L
aK
1−α
1−γ
V̇KY (t)
=
− Y
+α
+γ
−ρ + YaK + aN
1 + ηK i
VK (t)VK (t)(aK + aN ) 1 + ηN i

(B.4)

In the case in which VKY < VKY (t), it follows that limt→∞ VKY (t) = ∞. Therefore, thepath along which VKY (t) diverges to positive inﬁnity violates the TVC. Next, we consider
the case in which VKY > VKY (t). In this case, VKY (t) reaches zero in a ﬁnite period. As VKY (t)cannot take a negative value, the path along which VKY (t) approaches zero deviates fromfeasibility. Finally, we consider the path along which VKY = VKY (t). This case satisﬁes theTVC, limt→∞ e−ρt VKY [1+(aN /aK )] = 0, because the growth rate of e−ρt VKY (t)[1+(aN /aK )]is negative.
In the rest of this section, we check whether the BGP satisﬁes the other TVC:
lim e−ρt κ(t)m(t) = 0.

t→∞

(B.5)

On the BGP, from (26), (27) and (29), we obtain m(t) = [VK (t)/aK ](ηK aK gK + ηN aN gN ).
Using this, (A.9), (A.2) and (B.5), we obtain
lim e−ρtt→∞

VKY(ηK aK gK + ηN aN gN ) = 0.
aK

22

(B.6)

Therefore, we ﬁnd that the BGP satisﬁes the TVC; thus, the path initially jumping to
the BGP is the market equilibrium path.

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