ISSN (Print) 0473-453X

Discussion Paper No. 1118

ISSN (Online) 2435-0982

FINANCIAL DESTABILIZATION

Ken-ichi Hashimoto
Ryonghun Im
Takuma Kunieda
Akihia Shibata

February 2021

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

Financial Destabilization∗
Ken-ichi Hashimoto†Ryonghun Im‡

Takuma Kunieda§

Kobe University

Kyoto University

Kwansei Gakuin University

Akihisa Shibata¶
Kyoto University

February 3, 2021
Abstract
This paper uses a dynamic general equilibrium model to examine whether financial innovations destabilize an economy. Applying a neoclassical production
function, we demonstrate that as financial frictions are mitigated, the economy
loses stability and a flip bifurcation occurs at a certain level of financial frictions
under an empirically plausible elasticity of substitution between capital and labor. Furthermore, the amplitude of fluctuations increases as financial frictions
are mitigated and is maximized when the financial market approaches perfection. These outcomes imply that financial innovations are likely to destabilize
an economy.

Keywords: Financial innovations, endogenous business cycles, financial destabilization, heterogeneous agents.
JEL Classification Numbers: E13, E32, E44∗

This work has been financially supported by the Joint Usage/Research Center at ISER (Osaka
University) and KIER (Kyoto University), and the Japan Society for the Promotion of Science:
Grants-in-Aid for Scientific Research (Nos. 15H05728, 16H02016, 16H02026, 16K03624, 16K03685,
17H02516, 19K01638, 20H05631, 20K01631, and 20K01647) and Grant-in-Aid for JSPS Fellows(No. 18J01082). The authors would like to express thanks to Yoichi Gokan, Kazumichi Iwasa,
John Stachurski, Harald Uhlig, and Ping Wang for their invaluable comments to improve the paper.
The authors are also grateful to the participants at the JEA Fall Meeting in 2019 and the International Conference on Trade, Financial Integration and Macroeconomic Dynamics 2019 held at Kobe
University for their comments. All remaining errors, if any, are ours.
†
Associate Professor at Kobe University. Address: Graduate School of Economics, Kobe University, Rokko-dai 2-1, Kobe 657-8501, Japan; E-mail: hashimoto@econ.kobe-u.ac.jp‡
JSPS Research Fellow at Kyoto University. Address: Institute of Economic Research, Kyoto
University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan; E-mail: ryonghunim@gmail.com§
Corresponding author. Professor at Kwansei Gakuin University. Address: School of Economics,
Kwansei Gakuin University, 1-155 Uegahara Ichiban-cho, Nishinomiya 662-8501, Hyogo, Japan; Email: tkunieda@kwansei.ac.jp¶
Professor at Kyoto University. Address: Institute of Economic Research, Kyoto University,
Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan; E-mail: shibata@kier.kyoto-u.ac.jp1

1

Introduction

The development of modern monetary and financial systems began in the early 17th
century. In the financial development process, unceasing financial innovations haveenabled us to trade tremendous amounts of assets in the financial market. Under thesecircumstances, people have repeatedly witnessed large swings of boom-bust fluctuations, which are occasionally accompanied by credit market booms and collapses (seeBoissay et al., 2016). As such, researchers and policymakers often assert that financial innovations destabilize economies (e.g., Loayza and Ranciere, 2006; Guillaumont
Jeanneney and Kpodar, 2011; Rubio and Carrasco-Gallego, 2014). The purpose ofthis paper is to explore whether financial innovations destabilize an economy by applying a dynamic general equilibrium model with infinitely lived agents.
In our study, we assume that financial innovations directly mitigate financial frictions that entrepreneurs face. From the perspective of the 21st century, one may
imagine that this kind of financial innovation is caused by financial technology combined with information technology and/or artificial intelligence. In particular, due tothe development of financial technology in recent years, the amount of total debt andcredit in the private sector has ballooned far beyond the scale of gross domestic product in some countries. In this paper, we investigate how the mitigation of financial
frictions affects economic fluctuations.
The economy in our model is inhabited by three types of economic agents: firms,
workers, and entrepreneurs. The representative firm produces general goods from capital and labor with a neoclassical production technology that exhibits positive anddiminishing marginal products and is homogeneous of degree one with respect to both
capital and labor. Workers inelastically supply one unit of labor to the representative
firm to earn wage income in each period. They cannot borrow in the financial market, and their subjective discount factor is so small that their borrowing constraints
are always binding. Therefore, they consume all their earnings in each period, i.e.,they are hand-to-mouth consumers. Entrepreneurs receive idiosyncratic productivity
2

shocks in each period. Entrepreneurs who draw higher productivity shocks borrow in
the financial market and undertake an investment project to produce capital. Those
who draw lower productivity shocks store their wealth by lending in the financial market without engaging in an investment project. Because of idiosyncratic productivity
shocks, investors and lenders appear endogenously in each period. Entrepreneurs
face financial frictions, and they can borrow only up to a certain proportion of their
net wealth. The mitigation of financial frictions promotes capital accumulation because the allocation of production resources is improved, and thus macroeconomic
productivity increases.
The increase in the capital stock has two conflicting effects on capital income.
On the one hand, it places upward pressure on capital income because the sourceof capital income (i.e., the principal) becomes greater. On the other hand, it places
downward pressure on capital income because the decrease in the marginal productof capital lowers the interest rate. Therefore, the increase in the capital stock does
not necessarily raise capital income. In this paper, we assume that the elasticity ofthe marginal product of capital in the production technology increases as the capitalstock increases. In this case, at the early stage of capital accumulation, the positiveeffect of the increase in the capital stock on capital income surpasses the negative
effect, whereas the negative effect becomes stronger than the positive effect as capital
accumulates. Then, capital income increases at the early stage of capital accumulationand starts to decline when capital accumulation attains a certain threshold level. In
other words, the net effect that capital accumulation has on capital income has aninverted-U shape. In our model, this inverted-U shaped effect of capital accumulation
on capital income is crucial to producing endogenous business cycles.
To study how the mitigation of financial frictions affects economic fluctuations,
we produce bifurcation diagrams with respect to the extent of financial frictions.
Our findings are as follows. Under an empirically plausible elasticity of substitution
between capital and labor, as financial frictions are mitigated, the economy loses

3stability, and a flip bifurcation occurs at a certain level of financial frictions. Furthermore, the amplitude of business fluctuations increases as financial frictions are
mitigated and is maximized when the financial market approaches perfection. Therefore, financial innovations are likely to destabilize the economy.
The current paper is related to the literature on (deterministic) endogenous business cycles in the dynamic general equilibrium model, which have been theoreticallystudied by many researchers over the past thirty years and the plausibility of which
has recently been empirically supported by Beaudry et al. (2015, 2017, 2020). Examples of works that employ the overlapping generations model are Benhabib and Day(1982), Grandmont (1985), Farmer (1986), Reichlin (1986), Benhabib and Laroque(1988), Bertocchi and Wang (1995), Grandmont et al. (1998), and Rochon and Polemarchakis (2006). Whereas the models in all these works include outside money,
Yokoo (2000) demonstrates that endogenous business cycles can occur in an overlapping generations model without outside money, with the elasticity of the marginal
product of capital playing a crucial role in the occurrence of endogenous businesscycles, as in our model. Unlike Yokoo’s model, our model is inhabited by infinitely
lived agents, and we investigate the destabilization of an economy caused by financial innovations. Benhabib and Nishimura (1985), Boldrin and Denecker (1990), andNishimura and Yano (1995) employ an infinitely lived agent model in which endogenous business cycles are derived in an economy with two production sectors.
None of the abovementioned studies, however, explicitly consider financial frictions. Examples of papers that derive endogenous business cycles with financial frictions are Azariadis and Smith (1996, 1998), Favara (2012), Gokan (2011), Kikuchi(2008), Kikuchi and Stachurski (2009), Matsuyama (2007, 2013), Matsuyama et al.
(2016), and Myerson (2012, 2014). All these studies employ the overlapping generations model. In contrast, Woodford (1989) constructs a dynamic general equilibrium
model of infinitely lived capitalists and workers and derives endogenous business cycles. In his model, workers consume in a hand-to-mouth manner, not being allowed4

to borrow and lend in the financial market, and capitalists are homogeneous, so thatneither borrowing nor lending occurs among them. Although hand-to-mouth workers
are also assumed in our model, entrepreneurs can borrow up to the limit of borrowing
constraints as previously explained. In our model, the extent of financial frictions
plays a crucial role in the instability of the economy. Aghion et al. (2004) and Pintus(2011) also derive endogenous business cycles in a model of infinitely lived agents withfinancial frictions, but they assume a small open economy where the world interest
rate is exogenously given.1 Although Kunieda and Shibata (2017) investigate how
financial development affects endogenous business cycles in a dynamic general equilibrium model of infinitely lived agents, their economy does not exhibit endogenous
business cycles when the financial market approaches perfection. To the best of ourknowledge, no studies in the existing literature obtain the result that the amplitudeof endogenous business cycles increases as the financial market approaches perfection.
The remainder of this paper is organized as follows. In the following section, wedevelop the model and obtain the optimality conditions of entrepreneurs, workers, andthe representative firm. Section 3 derives an equilibrium in which a one-dimensional
dynamical system with respect to capital is obtained. In section 4, we investigate thepatterns of dynamic behaviors in the economy and observe the appearance of endogenous business cycles. In section 5, we find that the mitigation of financial frictions
destabilizes the economy. In this section, we also produce bifurcation diagrams withrespect to the extent of financial frictions. In section 6, we conclude the paper.

2

The model

A closed economy continues from period 0 to +∞ in discrete time indexed by t. Although the basic setting of financial frictions is similar to that of Kunieda and Shibata
1

Orgiazzi (2008) also investigates endogenous business cycles in a small open economy, extending
the model of Aghion et al. (2004), and demonstrates that the labor share plays an important rolein causing instability in the model of Aghion et al. (2004).

5

(2016), in which there are only entrepreneurs as active agents, it differs from theirmodel in that the economy in our model consists of a continuum of entrepreneurs,
identical workers, and a representative firm. The population of entrepreneurs is normalized to one, and that of workers is normalized to N . All these economic agentsare infinitely lived. The representative firm produces general goods from capital andlabor. Entrepreneurs are potential capital producers: in each period, those who draw
higher productivity shocks become capital producers (investors), and those who draw
lower productivity shocks become lenders.

2.1

Entrepreneurs

2.1.1

Timing of events

Consider the timing of events that a certain entrepreneur, say, entrepreneur j ∈ Xexperiences from period t to period t + 1, where X is the entire set of entrepreneurs.
At the beginning of period t, entrepreneur j has not yet received an idiosyncratic productivity shock for capital production. The consumption good market in period t is
opened at the beginning of the period and closed before an idiosyncratic productivity
shock is realized. Therefore, entrepreneur j makes a decision about consumption andsaving at the beginning of period t without knowing her productivity.
At the end of period t, there are two saving methods that entrepreneur j canapply: one is lending her savings in the financial market, and the other is initiating
an investment project. Because an idiosyncratic productivity shock is realized before
entrepreneur j determines her saving method, she optimally chooses between thesesaving methods while knowing her productivity. Lending one unit of savings in thefinancial market in period t yields a claim to rt+1 units of general goods in periodt + 1, where rt+1 is the gross real interest rate. Investing one unit of funds in a projectin period t yields Φjt units of capital in period t + 1, which is sold to the representative
firm at a price ρt+1 . In other words, entrepreneur j is endowed with an investmentjjtechnology such that kt+1
= Φjt · ijt , where kt+1
represents capital to be used for6general goods production in period t + 1, ijt is an investment, and Φjt is individualspecific productivity in capital production. Although Φjt is an idiosyncratic shock,the realization of low productivity cannot be insured against because no insurance
market for idiosyncratic productivity shocks exists. We assume that Φj0 , Φj1 , ... are
independent and identically distributed both over time and across entrepreneurs (thei.i.d assumption), and its cumulative distribution function is given by G(Φj ) the
support of which is given by [l, m] where l, m∈ R+ ∪ {+∞}. G(Φj ) is differentiable
and strictly increasing on the support.
2.1.2

Maximization problem

Entrepreneur j in period t maximizes the following expected lifetime utility:
Utj = Et"∞X#
β τ −t ln cjτ ,τ =twhere β ∈ (0, 1) is her subjective discount factor and cjτ is consumption in period τ .
The flow budget constraint of entrepreneur j is given byijτ + djτ = (ρτ Φjτ −1 ijτ −1 + rτ djτ −1 ) − cjτ for τ ≥ t,(1)

where djτ is lending if positive and borrowing if negative. In period t = 0, the flow
budget constraint is given by ij0 + dj0 = ρ0 k0 − cj0 , where k0 is the initial capitalendowment at birth and is common to all entrepreneurs.
In borrowing in the financial market, entrepreneur j faces a financial constraint,
which is given bydjτ ≥ −λajτ ,where ajτ := (ρτ Φjτ −1 ijτ −1 + rτ djτ −1 ) − cjτ for τ ≥ 1 (or aj0 := ρ0 k0 − cj0 for periodτ = 0) represents her savings.2 Henceforth, we call ajt entrepreneur j’s net worth.
2

Similar financial constraints are assumed in the literature (e.g., Aghion et al., 1999; Aghion and7

Note that λ ∈ (0, ∞) measures the extent of financial frictions: as λ increases becauseof financial innovations, the financial constraint is relaxed. The financial constraint
disables entrepreneur j from borrowing from the financial market more than λ times
her net worth. Because it follows from Eq. (1) that ijτ + djτ = ajτ , the financial
constraint is rewritten as
djτ ≥ −µijτ ,(2)where µ := λ/(1 + λ) ∈ (0, 1) also measures the extent of financial frictions. It isnecessary to impose the nonnegativity constraint of investment as follows:ijτ ≥ 0.

(3)

Entrepreneur j maximizes her expected lifetime utility Utj subject to Eqs. (1)-(3).
2.1.3

Optimal portfolio allocation

To consider an optimal portfolio allocation of entrepreneurs j’s net worth, we define
φt := rt+1 /ρt+1 .

(4)

If entrepreneur j draws her productivity such that Φjt > φt , she becomes an investor.
She optimally initiates an investment project with borrowing up to the limit of thefinancial constraint. If she draws her productivity such that Φjt ≤ φt , she lends her
entire net worth in the financial market to acquire the gross return rt+1 .3 Hence, φt
is the cutoff of the productivity shocks that divides entrepreneurs into investors andlenders in period t. Then, entrepreneur j’s optimal portfolio program is given by

ijt =




0

if Φjt ≤ φt



ajt
1−µ

if Φjt > φt ,Banerjee, 2005; Aghion et al., 2005; Antras and Caballero, 2009).
3
It is assumed that if entrepreneur j draws Φjt = φt , she becomes a lender.

8

(5)

and
djt =




ajt

if Φjt ≤ φt

 − µ aj

if Φjt > φt .

1−µ t

2.1.4

(6)

Euler equation

By defining Rτj := max{rτ , (ρτ Φjτ −1 − rτ µ)/(1 − µ)} and from Eqs. (5) and (6), we
can rewrite the flow budget constraint (1) as follows:ajτ = Rτj ajτ −1 − cjτ .

(7)

Entrepreneur j maximizes Utj subject to Eq. (7). The Euler equation for all t ≥ 0 is
given by"#11j
= βEt Rt+1 j
.
cjt
ct+1

2.2

(8)

Workers

Workers in our model are identical, being endowed with one unit of labor in eachperiod, and supply their labor to the production sector inelastically to earn wage
income. It is assumed that workers are hand-to-mouth consumers; that is, they
entirely consume their current labor income. Consider a certain worker, say, worker
h. Worker h’s consumption program can be given by
cht = wt ,(9)

for all t ≥ 0, where cht is worker h’s consumption.4
P∞
To derive Eq. (9), assume that workers’ lifetime utility is given by Uth = τ =t β̃ τ −t chτ where
workers’ subjective discount factor is so small that β̃ < 1/rt for all t ≥ 0 in equilibrium and workers
cannot borrow in the financial market. Then, workers consume all their labor income in each period.
King and Leape (1998) and Guiso et al. (2003) provide empirical evidence suggesting the existenceof hand-to-mouth consumers.
4

9

2.3

Production sector

The representative firm produces general goods with a production technology given by
Yt = F (Kt , Nt ), where Kt and Nt represent capital and labor in period t, respectively.
Capital depreciates entirely in one period. F (·, ·) is continuous and at least twice
differentiable with respect to Kt and Nt . It is assumed that limKt →0 F (Kt , Nt ) =limNt →0 F (Kt , Nt ) = 0. The production technology exhibits positive and diminishingmarginal products and constant returns to scale with respect to Kt and Nt . We definef (kt ) := F (Kt /Nt , 1) where kt := Kt /Nt is per worker capital. Then, it holds thatf 0 (kt ) > 0 > f 00 (kt ) with limkt →0 f (kt ) = 0.
Because the capital market is competitive, capital and labor are paid their marginal
products as follows:ρt = f 0 (kt ),(10)

wt = f (kt ) − f 0 (kt )kt .

(11)

The elasticity of the marginal product of capital is defined as follows:η(kt ) := −f 00 (kt )kt /f 0 (kt ) > 0.

(12)

For the sake of the following analysis, we prepare the following lemma.
Lemma 1. Suppose that t := −(dNt /dwt ) · (wt /Nt ) is the elasticity of demand forlabor. Then it holds that

 
wt
1
η(kt ) =
.
t
ρt kt

(13)

Proof. See the Appendix.
Eq. (13) expresses the relationship among wage, capital income, the elasticity of themarginal product of capital and the elasticity of demand for labor.

10

3Equilibrium

A competitive equilibrium is given by sequences of prices {wt , ρt , rt+1 } for all t ≥ 0
and allocation {cjt , ajt , ijt , djt } for all j and t ≥ 0, {cht } for all h and t ≥ 0, and {kt }for all t ≥ 0, such that (i) for each j, entrepreneur j maximizes her lifetime utility
from each period t onward; (ii) for each h, worker h consumes so that cht = wt ineach period; (iii) the representative firm maximizes its profits in each period; and (iv)the financial market (in section 3.2), the capital market (in section 3.3), the general
goods market, and the labor market (Nt = N ) all clear.
j
From Eq. (7), it follows that Et [ajt+1 /cjt+1 ]=ajt Et [Rt+1/cjt+1 ] − 1. Substituting theEuler equation (8) into this equation, we obtain ajt /cjt =βEt [ajt+1 /cjt+1 ] + β. From thisequation and the law of iterated expectations, it follows that ajt /cjt =β s Et [ajt+s /cjt+s ]+β+
β 2 +...+β s . Since the transversality condition is given by lims→∞ β s Et [ajt+s /cjt+s ] = 0,
it follows that ajt /cjt = β/(1 − β) for all t ≥ 0. Substituting this equation into Eq.(7)yieldsajt = βRtj ajt−1 .

3.1

(14)

Aggregate net worth

Define Ξt =

j ∈ X : Φjt ≤ φt . Then, Ξt is the set of entrepreneurs who draw

individual-specific productivity shocks such that Φjt ≤ φt in period t. From Eq.
(14), it follows thatZAt :=ajt dj =j∈X

Z

βRtj ajt−1 dj.

(15)

j∈X

The aggregate income across all entrepreneurs is equal to the sum of the total capital
R
R
income, and thus, j∈X Rtj ajt−1 dj = ρt Kt where Kt = j∈X\Ξt Φjt−1 ijt−1 dj is the total
capital used for general goods production. Therefore, the aggregate net worth acrossall entrepreneurs is obtained as follows:
At = βρt Kt .
11

(16)

3.2

Financial market clearing condition

The financial market clearing condition is given byZ

djt dj = 0.

(17)

j∈X

Substituting Eq. (6) into Eq. (17) yields
Z

µ
1−µ

ajt dj −j∈ΞtZ

ajt dj = 0.

(18)

ajt dj = G(φt )At(19)

ajt dj = (1 − G(φt ))At ,(20)

j∈X\Ξt

The i.i.d. assumption computesZ

ajt dj =

j∈Ξt

Z

Z
dj ·

j∈Ξt

j∈X

and
Z

ajt dj =dj ·j∈X\Ξtj∈X\Ξtwhere we have used

R

Z

Z

dj =
j∈Ξt

R

Φjt ≤φtj∈XdG(Φjt ) = G(φt ) for the second equalities in Eqs.

(19) and (20). Substituting Eqs. (19) and (20) with Eq. (16) into Eq. (18) yields
G(φt )At −µ(1 − G(φt ))At = 0.
1−µ(21)Eq. (21) yields G(φ∗ ) = µ, from which an equilibrium cutoff can be uniquely determined as follows:φ∗ = G−1 (µ) =: φ∗ (µ).

(22)

Lemma 2 below shows that φ∗ strictly increases with µ.
Lemma 2. dφ∗ (µ)/dµ > 0.
Proof. G(·) is differentiable and strictly increasing, and µ ∈ (0, 1). Hence, φ∗ isuniquely determined and strictly increases with µ.

12

3.3

Capital market clearing condition

The capital market clearing condition is given byZΦjt · ijt dj.

Kt+1 =(23)j∈X\ΞtBy using Eq. (5), the i.i.d. assumption rewrites Eq. (23) as1
Kt+1 =1−µZ

Φjt · ajt dj =j∈X\Ξt11−µZ
Φjt >φt

Φjt dG(Φjt ) ·Z

ajt dj.

(24)

j∈X

Substituting Eqs. (16) and (22) into Eq. (24) yields
Kt+1 =
where H(φ∗ ) :=R

Φjt >φ∗βH(φ∗ )
ρt Kt ,
1−µ(25)

Φjt dG(Φjt ). One can demonstrate that limµ→1 H(φ∗ )/(1 − µ) =m by applying L’Hopital’s rule and the inverse function theorem with Eq. (22),provided that m (which is the highest productivity) is a finite value. Therefore, theeconomy is well defined when the financial market approaches perfection. In thiscase, aggregate saving in the economy is used by the most productive entrepreneurs
for capital production.

4

Dynamics

Dividing both sides of Eq. (25) by N and inserting Eqs. (10) and (22) into theresulting equation, we obtain a dynamic equation of per worker capital as follows:
kt+1 = Ω(µ)f 0 (kt )kt =: Ψ(kt ; µ),where Ω(µ) := βH(φ∗ (µ))/(1 − µ).
Lemma 3. ∂Ω(µ)/∂µ > 0.
13

(26)

Proof. See the Appendix.
From Eq. (26) and Lemma 3, it follows that ∂Ψ(kt ; µ)/∂µ > 0. This implies thatthe configuration of Eq. (26) shifts up as financial frictions are mitigated. This isbecause as financial frictions are mitigated, low-productivity entrepreneurs are ruledout of capital production, and thus, productivity in the economy as a whole becomes
high.

4.1

Steady state

Suppose that k ∗ is the nontrivial steady-state value of capital. From Eq. (26), k ∗satisfies the following equation:
1 = Ω(µ)f 0 (k ∗ ).

(27)

Assuming the existence of the nontrivial steady-state value of capital for any µ∈ (0, 1),
one notes from Eq. (27) that k ∗ has a one-to-one relationship with µ because ofLemma 3 and can be written in terms of µ as k ∗ = f 0 −1 (1/Ω(µ)) =: k ∗ (µ). In
particular, we obtain the following proposition.
Proposition 1. The steady-state value of per worker capital strictly increases asfinancial frictions are mitigated, i.e., ∂k ∗ (µ)/∂µ > 0.
Proof. See the Appendix.
The mitigation of financial frictions promotes the accumulation of capital stock in thesteady state because productivity in the economy becomes high and output increases
as financial frictions are mitigated.

14

4.2

Dynamic patterns

In this subsection, we investigate the dynamic behavior of the capital stock that ischaracterized by Eq. (26). From Eq. (26), it follows that
Ψ0 (kt ; µ) = Ω(µ)[f 0 (kt )kt ]0 = Ω(µ)f 0 (kt )[1 − η(kt )],(28)

where η(kt ) is the elasticity of the marginal product defined by (12). We prepare
the following assumption with respect to η(kt ) to explore the case in which we areinterested.
Assumption 1. η 0 (kt ) > 0, limkt →0 η(kt ) < 1, and limkt →∞ η(kt ) > 2.
Under Assumption 1, there exist unique values of k̂ and k̃ in (0, ∞) such that η(k̂) = 1
and η(k̃) = 2. Then, it follows that
∈ (0, 1]for 0 < kt ≤ k̂η(kt )∈ (1, 2]for k̂ < kt ≤ k̃ ∈ (2, ∞) for k̃ < k .
t(29)Assumption 1 implies that the elasticity of the marginal product strictly increases with
per worker capital. This assumption holds with a production function exhibiting a1

constant elasticity of substitution such as F (Kt , N ) = (γ1 N −σ + γ2 Kt−σ )− σ undersome parameter conditions, which is used for the numerical analysis in section 5.2.
4.2.1

Local dynamics

Eq. (26) can be linearized in the neighborhood of the steady state as kt+1 − k ∗ =Ψ0 (k ∗ ; µ)(kt − k ∗ ), which can be rewritten as
kt+1 − k ∗ = [1 − η(k ∗ )](kt − k ∗ )

15

(30)

by using Eqs. (27) and (28). The local stability in the neighborhood of the steadystate immediately follows from Eqs. (29) and (30), as summarized in Proposition 2
below.
Proposition 2. Suppose that Assumption 1 holds. Then, if 0 < k ∗ < k̃, the steadystate of Eq. (26) is stable, and if k ∗ > k̃, the steady state of Eq. (26) is unstable.
Proof. From Eqs. (29) and (30), it follows that |1 − η(k ∗ )| < 1 if 0 < k ∗ < k̃, and1 − η(k ∗ ) < −1 if k ∗ > k̃. Then, the claims hold.

4.2.2

Global dynamics

The configuration of Ψ(kt ; µ) depends upon the sign of Ψ0 (kt ; µ) with µ given. Since
the sign of Ψ0 (kt ; µ) is the same as that of 1 − η(kt ), as seen in Eq. (28), we have alemma regarding the configuration of Ψ(kt ) below.
Lemma 4. Suppose that Assumption 1 holds. Then, Ψ(·; µ) is a single-peaked mapping; i.e., Ψ(kt ; µ) increases with kt ∈ (0, k̂) and decreases with kt ∈ (k̂, ∞).
Proof. Since the sign of Ψ0 (kt ; µ) is the same as that of 1 − η(kt ), if 0 < kt < k̂, itholds that Ψ0 (kt ; µ) > 0, and if kt > k̂, it holds that Ψ0 (kt ; µ) < 0, which leads to adesired conclusion.
16

To discuss the global dynamics of Eq. (26), we draw typical configurations of Eq.
(26) in Figure 1. As seen in Panel A, if the steady-state capital k ∗ is smaller than k̂,it holds that 0 < 1 − η(k ∗ ) < 1. In this case, the economy monotonically converges tothe steady state if k0 < k ∗ . If the steady-state capital k ∗ is greater than k̂ and smaller
than k̃, the steady state is stable and converges to the steady state with oscillation,
as seen in Panel B. If the steady-state capital stock k ∗ is greater than k̃, the steadystate is unstable, as seen in Panel C, and the economy exhibits endogenous businesscycles.

5

Financial destabilization

In this section, we demonstrate that financial innovations can destabilize the economy.

17

5.1

Bifurcation

The steady-state capital stock k ∗ increases with µ, as see in Proposition 1, whereask̃ does not vary with µ. To investigate bifurcations, the following assumption is
imposed:Assumption 2. limµ→0 k ∗ (µ) < k̃ < limµ→1 k ∗ (µ).
1Again, the use of F (Kt , N ) = (γ1 N −σ + γ2 Kt−σ )− σ satisfies Assumption 2 under someparameter conditions. Under Assumptions 1 and 2, there exists a unique µ̃ in (0, 1)
such that k ∗ (µ̃) = k̃. Then, we obtain Proposition 3 below.
Proposition 3. Suppose that Assumptions 1 and 2 hold. Suppose also that µ increases from 0 to 1. Then, a flip bifurcation occurs in the steady state at µ = µ̃.
Proof. Since η(k ∗ (µ̃)) = 2, it follows that 1 − η(k ∗ (µ̃)) = −1, and from Proposition 1
and Assumption 1, we have ∂(1 − η(k ∗ (µ̃))/∂µ = −η 0 (k ∗ (µ̃))(∂k ∗ (µ̃)/∂µ) < 0. Then,a flip bifurcation occurs at µ = µ̃.
The intuition regarding the occurrence of the flip bifurcation is as follows. If
entrepreneurs earn higher income than that in the steady state, they invest morethan the amount in the steady state. Then, the increased capital has two conflicting
effects on capital income ρt kt . On the one hand, the increase in kt places upward
pressure on capital income because the source of capital income becomes greater.
On the other hand, the increase in kt decreases the marginal product of capital ρt ,which has negative impacts on capital income. When the negative effect dominates
the positive one, the larger investment yields a very low return on capital incomein the next period. Because of lower earnings, entrepreneurs now invest less than
the amount in the steady state, and thus earn more than in the steady state in thenext period. These fluctuations around the steady state become unstable as financial
frictions are mitigated. One notes that when µ < µ̃, the steady state is stable, andthe economy converges to the steady state. As µ increases from 0 to 1, Ψ(kt ; µ)shifts up; as seen in Figure 2, a flip bifurcation occurs in the steady state, and the18economy loses stability when µ = µ̃. If µ becomes greater than µ̃, or equivalently,
if η(k ∗ ) > 2, the steady state becomes locally unstable, and the economy exhibits
endogenous business cycles, as illustrated in Panel C in Figure 1. This means that
financial innovations can destabilize the economy.
The condition for the steady state to become locally unstable can be rewritten interms of labor and capital income and the elasticity of labor demand by using Lemma
1. Lemma 1 rewrites η(k ∗ ) > 2 as follows:  
1
w̃
> 2,
˜
ρ̃k̃

(31)

where ˜ is the value of t evaluated at kt = k̃. The condition derived in Woodford(1989) for local instability of the steady state can be reduced to Eq. (31) provided thatlabor is inelastically supplied in his model. In contrast to our model, both capitalists
and workers are identical in his model, and thus, neither borrowing nor lending occurs
between the two groups and within each group. The distinctive features in our model
relative to Woodford’s model are the heterogeneity in entrepreneurs’ productivity andthe financial friction that they face. The heterogeneity in entrepreneurs’ productivity
yields the situation in which borrowing and lending occur among entrepreneurs andthe financial friction allows us to investigate how the mitigation of financial frictions
destabilizes the economy.

5.2

Numerical analysis

This subsection provides a numerical example. We produce bifurcation diagrams withrespect to µ to examine how the mitigation of financial frictions affects qualitative
characteristics of dynamic behaviors in the economy.

19

5.2.1

Specification and parameterization

The functional form of the production technology is specified as
1

F (Kt , N ) = (γ1 N −σ + γ2 Kt−σ )− σ ,which exhibits a constant elasticity of substitution. F (Kt , N ) can be rewritten as aper worker production function as follows:1f (kt ) = (γ1 + γ2 kt−σ )− σ ,(32)where γ1 , γ2 ∈ (0, 1) and γ1 + γ2 = 1. From Eq. (32), we have the elasticity of themarginal product of capital as follows:−f 00 (kt )kt
η(kt ) == (1 + σ)
f 0 (kt )γ1
γ1 + γ2 kt−σ


.

(33)

We focus on the case in which σ > 0, so that Assumption 1 can hold and limKt →0 F (Kt , N ) =limN →0 F (Kt , N ) = limkt →0 f (kt ) = 0 can hold. In this case, k̂ can be computed as11k̂ = [γ2 /(σγ1 )] σ , and k̃ can be computed as k̃ = [2γ2 /(γ1 (σ − 1))] σ only when σ > 1,
where, as previously defined, k̂ and k̃ satisfy η(k̂) = 1 and η(k̃) = 2, respectively.
1+σFrom Eq. (26) and f 0 (kt )kt = γ2 kt /(γ1 ktσ + γ2 ) σ , the dynamic equation for kt isobtained as
kt+1 = Ψ(kt ; µ) := Ω(φ∗ )

γ2 kt
1+σ

(γ1 ktσ + γ2 ) σ

.

(34)

In Eq. (34), one notes that Ψ(0; µ) = 0 for kt = 0 and Ψ(kt ; µ) > 0 for kt > 0. It
follows from Eq. (34) that

Ψ0 (kt ; µ) = Ω(φ∗ )

γ1 γ2 σ(γ1 ktσ + γ2 )

1+2σ
σ

20





≥ 0 for kt ≤ k̂
γ2
− ktσ

σγ1
< 0 for kt > k̂,(35)which implies that Ψ(kt ; µ) increases with kt ∈ (0, k̂) and decreases with kt ∈ (k̂, ∞).
Thus, Ψ(·; µ) is a single-peaked mapping. We impose a parameter condition such that−1/σΩ(φ∗ )γ2> 1 can hold. Under this parameter condition, Eq. (34) has a nontrivial−1/σ

steady state, k ∗ , because Ψ0 (0) > 1 ⇐⇒ Ω(φ∗ )γ2> 1.

It is assumed that Φjt follows a uniform distribution over [0, m]. Therefore, we haveG(φ) = φ/m and H(φ) = (m2 − φ2 )/(2m).5 Under the assumption of the uniform
distribution, the productivity cutoff is given byφ∗ = µm.

(36)

In the numerical analysis, bifurcation diagrams are produced by varying µ. We set
β = 0.96 following standard real business cycles theory and γ1 = 0.67 and γ2 = 0.33
taking into account the standard income share ratio between labor and capital. We set
a relatively large value for m, that is, m = 10. In this case, the average productivityof entrepreneurs is equal to H(0) = 5. Under these parameter settings, we examine
the three cases of σ = 1, 1.46 and 2.5.6
5.2.2

Bifurcation diagrams

The bifurcation diagrams in Figure 3 depict the effects that financial constraints haveon the dynamics of capital in equilibrium.
Recent empirical studies report that the elasticity of substitution between capitaland labor takes a value from 0.4 to 0.6 (e.g., Klump et al., 2007; Chirinko, 2008;
León-Ledesma et al., 2010). Then, we examine three cases in which the elasticity of
substitution is approximately equal to 0.50, 0.41, and 0.29. As seen in the figure, whenσ = 1 (the elasticity of substitution between capital and labor is 1/(1 + σ) = 0.50),
the economy monotonically converges to a steady state for any value of µ because theIn the Appendix, we also examine the case in which Φjt follows a Pareto distribution for arobustness check and obtain a similar result to that obtained from the uniform distribution.
−1/σ
6
Under these parameter settings, it holds that Ω(φ∗ )γ2> 1 for σ = 1, 1.46 and 2.5 andµ ∈ (0, 1), and thus, Eq. (34) has a nontrivial steady state, k ∗ .
521

steady state of Eq. (34) is stable. The capital stock in the steady state increases as µincreases because the mitigation of financial frictions promotes capital accumulation.
When σ = 1.46 (the elasticity of substitution between capital and labor is equal to1/(1 + σ) ≈ 0.41), a flip bifurcation occurs at a certain value of µ and a periodtwo cycle arises. One notes that as µ increases, the amplitude of the period-two
cycle increases. When σ = 2.5 (the elasticity of substitution between capital andlabor 1/(1 + σ) ≈ 0.29), complex dynamic behaviors are obtained. In this case,whereas smaller values of µ produce a period-four cycle, period-doubling bifurcations
repeatedly occur as µ increases. Eventually, greater values of µ produce complex
dynamics in the economy. As in the case of σ = 1.46, the amplitude of fluctuations
increases as µ increases.

In summary, in the case in which endogenous business cycles occur, the amplitudeof fluctuations increases, and this amplitude is maximized when the financial market
approaches perfection. This outcome implies that the mitigation of financial frictions
is likely to destabilize the economy.

6

Conclusion

Do financial innovations destabilize economies? Many researchers and policymakers
have argued that financial innovations destabilize economies. To address this issue, we22have developed a simple dynamic general equilibrium model with financial frictions.
Applying the production function in which the elasticity of the marginal productof capital increases as capital accumulates, we investigate the characteristics of the
dynamic behavior of the economy. According to recent empirical studies, the elasticity
of substitution between capital and labor ranges from 0.4 to 0.6. Our numericalanalysis shows that when it is 0.41, a flip bifurcation occurs at an intermediate extent
of financial frictions, and a period-two cycle arises. Furthermore, our analysis showsthat as financial frictions are mitigated, the amplitude of the cycle increases. These
outcomes imply that financial innovations can destabilize economies. Our model canbe extended to introduce intrinsically useless assets. The investigation into how thepresence of such assets changes the destabilization property is left for future research.

Appendix
Proof of Lemma 1
From (11), we have dwt /dNt = −f 00 (kt )kt2 /Nt . Then, it follows that−f 00 (kt )kt
f 0 (kt )
!

dwt
Nt dN
kt
t
=
kt2
f 0 (kt )!1wt= wt dNt
f 0 (kt )kt
Nt dwt
 

1
wt
=
.
t
ρt kt

η(kt ) =23(37)

Proof of Lemma 3
From Ω(µ) = βH(φ∗ (µ))/(1 − µ), it follows that∂Ω(φ∗ )
=β∂µ∗(1 − µ) + H(φ∗ )−φ∗ G0 (φ∗ ) dφdµ!
.

(1 − µ)2(A.1)

Since G(φ∗ ) = µ and G0 (φ∗ )(dφ∗ /dµ) = 1, Eq. (A.1) can be rewritten as∂Ω(φ∗ )=β∂µH(φ∗ ) − φ∗ (1 − G(φ∗ ))(1 − µ)2


.

(A.2)

In Eq. (A.2), it holds that H(φ∗ ) − φ∗ (1 − G(φ∗ )) > 0. Therefore, ∂Ω(φ∗ )/∂µ > 0.

Proof of Proposition 1
From Eq. (27), it follows thatln Ω(µ) = − ln f 0 (k ∗ ).

(A.3)

Differentiating Eq. (A.3) with respect to µ yields


1
Ω(µ)



∂Ω(µ)
∂µ




=

f 00 (k ∗ )− 0 ∗f (k )



∂k ∗
∂µ


,

or equivalently,∂k ∗f 0 (k ∗ )=∂µ−f 00 (k ∗ )Ω(µ)



∂Ω(µ)
∂µ


(A.4)

In Eq. (A.4), it follows from Lemma 3 that ∂Ω(µ)/∂µ > 0. Therefore, ∂k ∗ (µ)/∂µ > 0.

Bifurcation diagrams in the case of the Pareto distribution
Suppose that Φjt follows a Pareto distribution such that24

G(φ) = n
 1− αif α < φ 0

if φ ≤ α,φwhere α > 0 and n ≥ 2. Then, we have H(φ) = φ[1 − G(φ)]n/(n − 1). It is
straightforward to obtain the productivity cutoffs such that
φ∗ =

α
1

(1 − µ) n

.

(A.5)

To produce bifurcation diagrams, we set α = 1 and n = 2 with other parametervalues remaining the same as those in subsection 5.3. From Eq. (A.5), we obtain1Ω(φ∗ ) = βH(φ∗ (µ))/(1 − µ) = αn/[(1 − µ) 2 (n − 1)]. As seen in this equation, Ω(φ∗ ) is
not well defined when µ → 1 because m is not finite in the case of Pareto distributions.
Then, we impose the upper limit of µ at µ = 0.99 when producing the bifurcation
diagrams. Figure A shows the effects of financial constraints on the dynamic behaviorof capital. The results are basically the same as those of the uniform distribution.

25

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